Understanding Trinity, is the belief that each Person has independent Will?

In your question you appear to be overlooking two very important things.

1. Jesus was not only God, but he was also human.

2. Humans feel physical pain, even though Spirits do not.

In the Scriptures that you cite it must be remembered that Jesus was in an inordinate state. That is to say that the Deity Jesus was aware of the oncoming pain:

Matthew 17:22 and 23 And while they abode in Galilee, Jesus said unto them, The Son of man shall be betrayed into the hands of men: 23 And they shall kill him, and the third day he shall be raised again. And they were exceeding sorry.

and as a man he would know just how painful that was going to be.

The plea to the father would have to come from the human part of Jesus, and it is not surprising that the man Jesus' will was not to go through all that torture and pain.

Luke 22:41 through 44 And he was withdrawn from them about a stone's cast, and kneeled down, and prayed, 42 Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done. 43 And there appeared an angel unto him from heaven, strengthening him. 44 And being in an agony he prayed more earnestly: and his sweat was as it were great drops of blood falling down to the ground.

This is a physical human phenomenon, and Medical Doctors have witnessed blood emitting from pores of people under extreme anxiety. Jesus would have had to fit into that category; having the foreknowledge of his torture and death on the cross.

As far as:

Matthew 27:46 And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?

God cannot look upon sin, and when Jesus became sinful by assuming our sins, he was forsaken by God. and it is imperative that we understand that God was forsaking the man Jesus on the cross, and that means all three of the Trinity were abandoning his physical man.

I know that there are going to be many comments concerning how Jesus could abandon himself, but that is not the case. The eternal son which was present at creation abandoned the physical body it had been inhabiting while on earth. consider:

John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.

Hope this helps.

Edit: removed old answer

I think this link will help explain what you are asking: Was Jesus Limited While On Earth?

For future reference if that site or page disappears from the internet it is quoted below:

Was Jesus Limited While On Earth?

Wednesday, January 29th, 2014    tags:The Holy Spirit,Theology

Q. I have a question about Our Lord when He was on earth. I have heard it said that His knowledge or power was limited on earth according to God the Father’s discretion. That is why Jesus frequently prayed to the Father or the scriptures say, “Not even the Son of Man knows” about a particular thing. Do you agree with this?

A. I believe when Jesus agreed to become a man, He voluntarily set aside His Godly powers and limited Himself to the powers available to mankind. This is why He attributed His ability to perform miracles to the Holy Spirit (Matt. 12:28), was in daily prayer with the Father, claimed not to know certain things (Matt. 24:36), and promised us we could do even greater things than he had done (John 14:12).

There are at least 3 reasons why we know this is true. First, God can’t be tempted (James 1:13), but Jesus was tempted in every way and yet was without sin. (Hebr. 4:15) Second, according to the Law of Redemption He had to become a man to redeem what Adam had lost (Lev. 25:25) and save us, and third He had to become a man so He could be put to death. It’s impossible to kill God.

The question, as it stands, really isn't soluable.

Reason #1: The Crucifixion raises other Trinitarian questions

First and foremost, the Trinity itself is hard enough to understand. There is no good analogy and any attempt to make one will necessarily fail by over emphasizing oneness or threeness.

Worse, the Crucifixion and death of Christ means that either Jesus was in some fashion separated from the Father were separated on the cross, or else there is the problem of Patripassianism - meaning that the Father somehow dies too.

One solution kenosis modifies the hypostatic union to explain how Jesus can be fully God and yet somehow not able to do everything God did, but it is just one solution.

Reason #2: The Trinity logically requires that the Wills cannot be separated

In 2007, William P. Young write a book called The Shack which was rather popular in the Contemporary Christian market for a bit. The story itself imagined a Mack, a man whose daughter was killed, spending a weekend in which he got to meet Jesus face to face. Along the way, develops the characters of "Papa" an African-American female personification of God the Father, Jesus, and Sarayu, the Holy Spirit. His general idea was to understand the Trinity as the God whose very essence modeled true love-in-relationship that mankind was supposed to be.

From that vantage point, he was able to illustrate the Trinity and its coordinate implications for separate wills that I, as an evangelical, found useful.

To begin with, Young acknowledges that man has only a finite capacity to comprehend a finite God. He has "Papa" say:

"The problem is that many folks try to grasp some sense of who I am by taking the best version of themselves, projecting that to the nth degree, factoring in all the goodness they can perceive, which often isn’t much, and then call that God. And while it may seem like a noble effort, the truth is that it falls pitifully short of who I really am. I’m not merely the best version of you that you can think of. I am far more than that, above and beyond all that you can ask or think.”

“Never mind that,” she continued. “What’s important is this: If I were simply One God and only One Person, then you would find yourself in this Creation without something wonderful, without something essential even. And I would be utterly other than I am.”

“And we would be without . . . ?” Mack didn’t even know how to finish the question.

“Love and relationship. All love and relationship is possible for you only because it already exists within Me, within God myself. Love is not the limitation; love is the flying. I am love.”

Mack and "Papa" explicitly mention the Trinity, and Mack's inability to understand, and then get to the question you have: Namely, what happens if "Papa" and "Jesus" and "Sarayu" are in disagreement. Sarayu picks up the conversation, saying:

“Mackenzie, we have no concept of final authority among us, only unity. We are in a circle of relationship, not a chain of command or ‘great chain of being’ as your ancestors termed it. What you’re seeing here is relationship without any overlay of power. We don’t need power over the other because we are always looking out for the best. Hierarchy would make no sense among us. Actually, this is your problem, not ours.”

“Really? How so?”

“Humans are so lost and damaged that to you it is almost incomprehensible that people could work or live together without someone being in charge.”

“It’s one reason why experiencing true relationship is so difficult for you,” Jesus added. “Once you have a hierarchy you need rules to protect and administer it, and then you need law and the enforcement of the rules, and you end up with some kind of chain of command or a system of order that destroys relationship rather than promotes it. You rarely see or experience relationship apart from power. Hierarchy imposes laws and rules and you end up missing the wonder of relationship that we intended for you.”

Sarayu continued, “When you chose independence over relationship, you became a danger to each other. Others became objects to be manipulated or managed for your own happiness. Authority, as you usually think of it, is merely the excuse the strong use to make others conform to what they want.”

...

“We carefully respect your choices, so we work within your systems even while we seek to free you from them,” Papa continued. “Creation has been taken down a very different path than we desired. In your world the value of the individual is constantly weighed against the survival of the system, whether political, economic, social, or religious—any system actually. First one person, and then a few, and finally even many are easily sacrificed for the good and ongoing existence of that system. In one form or another this lies behind every struggle for power, every prejudice, every war, and every abuse of relationship. The ‘will to power and independence’ has become so ubiquitous that it is now considered normal.”

“It is the human paradigm,” added Papa, having returned with more food. “It is like water to fish, so prevalent that it goes unseen and unquestioned. It is the matrix; a diabolical scheme in which you are hopelessly trapped even while completely unaware of its existence.”

Jesus picked up the conversation. “As the crowning glory of Creation, you were made in our image, unencumbered by structure and free to simply ‘be’ in relationship with me and one another. If you had truly learned to regard each other’s concerns as significant as your own, there would be no need for hierarchy.”

Mack sat back in his chair, staggered by the implications of what he was hearing. “So are you telling me that whenever we humans protect ourselves with power . . .”

“You are yielding to the matrix, not to us,” finished Jesus.

“And now,” Sarayu interjected, “we have come full circle, back to one of my initial statements: You humans are so lost and damaged that to you it is almost incomprehensible that relationship could exist apart from hierarchy. So you think that God must relate inside a hierarchy like you do. But we do not.”

In short, Young is suggesting that the question is most properly answered "Mu", meaning the question has a certain level of sense, but that a direct answer is ultimately non-sensical because the question holds an inherent problem within it.

Here, it is most fair to represent Young as saying that the Trinity is in relationship but not in hierarchy. The question of a separate "will" is sensical, but ultimately not answerable, because the relationship is so tight. (A situation that in statistics would be called Multicollinearity).

Put yet another way, there is no way to separate the wills, but that does not abrogate their theoretical existence. God's perfection in relationship means that wills cannot be in contradiction to one another, but this does not mean that they are the same.

Or, if the 8th Century Theologian Boethius in De Trinitae can better say:

Thus ‘different’ is said in respect to either genus, species or number. But it is variety among accidents that produces difference in respect to number. For three men differ nei- ther in genus nor species, but in their accidents; for even if we mentally separate all acci- dents from them, there is still a different <168.60> location for each and all of them, which we can in no way imagine to be one: for two bodies will not occupy one location; and location is an accident. Therefore these three men are many in respect to number, since they become many by their accidents.

Reason #3: Second Council of Constantinople (553 AD) is hard to understand

Really, this is your full answer. It will go into the nature of the hypostatic union, the h*m*ousion, and the definition of persons, wills and essences. This will literally take semesters of study to be comprehensible, but is your official "Christian" answer.

Ultimately, the creeds are trying to avoid errors such Modalistic Monarchism and still make it comprehensible. Good luck.

Catholic teaching and understanding is that Christ has two wills, divine (of which, there is only one¹) and human - without the human, to my understanding, he couldn't have redeemed in the manner he redeemed [cf. Heb 5:8 & Phil 2:7-9].

^{1. In the Godhead the essence, will, and action are but one. - cf. The divine unity in The Blessed Trinity | New Advent.}

From the catechism of the Catholic Church,

Christ's human will CCC 475 Similarly, at the sixth ecumenical council, Constantinople III in 681, the Church confessed that Christ possesses two wills and two natural operations, divine and human. They are not opposed to each other, but cooperate in such a way that the Word made flesh willed humanly in obedience to his Father all that he had decided divinely with the Father and the Holy Spirit for our salvation.² Christ's human will "does not resist or oppose but rather submits to his divine and almighty will."³

<sup>2. cf. Council of Constantinople III (681): DS 556-559.
3. Council of Constantinople III: DS 556.</sup>

Please see also On the human soul of Christ in The Incarnation | New Advent.

As for “My God, my God, why hast thou forsaken me?”, the best explanation I have read is that Jesus said these words aloud, and continued to pray Psalm 22 silently. It ends in vindication and victory.

Matt 27:46 (RSVCE) 46 And about the ninth hour Jesus cried with a loud voice, “Eli, Eli, la′ma sabach-tha′ni?” that is, “My God, my God, why hast thou forsaken me?”^[a]

^{Footnotes:
a. Jesus applies Psalm 22 (Vulgate 21) to himself.}