What was Jesus' soul?

A soul is a the union of one's spirit and body. A disembodied spirit, such as a spirit before birth or after death does not have a body and is therefore not a soul. A lifeless body, one that is dead, is not a soul but is merely a corpse.

In John 12:27, Jesus is prophesying concerning His own death, for He knows that it is only a few days away and He also knows that the sacrifice He will make will descend below all things and be the worst physical, spiritual, and emotional pain that anyone had ever endured. But only he could do it, because he was a half-God, half-man. He had God the Father as His Biological Father and a mortal Mary for His mother. Therefore, He had power over death, but was still capable of dying. No man could take away His life, but He could give up His life. He was nevertheless subject to every physical hardship of mortals including pain, sickness, grief, sorrow, temptations, toil, labor, and heart aches.

In the cases where Jesus is referring to His soul being sorrowful, his body and spirit are pained both physically and spiritually because this marks the beginning of Christ's sacrifice for all men everywhere. He is taking upon himself the sins of the entire world and of every person who had ever lived or would live.

This moment was the most critical moment in eternity. Jesus had a choice, He could either continue and drink "this cup" and yet he would be saved, for he was the only sinless being who had ever lived, and he could let all mankind to be lost, or he could drink "the cup" and finish the requisite sacrifice that would allow all men the possibility to live with God in heaven.

Luke records of this, "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." (Luke22:44) There in the garden called Gethsemane (meaning Olive Press), the Son of God, the Prince of Peace, the Savior of the World, the Promised Messiah, knelt and bowed under the crushing weight of the sins of the world as He took upon Himself the sins of all mankind that he might redeem all men, if they would choose Him and cast their sins upon him.

Following this ordeal, which He had asked His disciples to stay awake and witness, he was betrayed by Judas, arrested, and the abusive Jewish leaders began the trials that would result in His death.

Jesus was well aware of all that had happened and all that would yet take place. Is it any wonder that His soul was "exceeding sorrowful, even unto death"?

A body is animated by the soul. Jesus’ body was no different, except that it was both fully human and fully divine: there wasn’t a “human part” and a “divine part”. As his whole nature was fully human and fully divine, it follows that his soul was both fully human and fully divine as well, for he could not be the Word Incarnate without the Word being part of all of his incarnate nature.

This is attested by the Fathers, for example St John Damascene:

By the fact that at Christ's death his soul was separated from his flesh, his one person is not itself divided into two persons; for the human body and soul of Christ have existed in the same way from the beginning of his earthly existence, in the divine person of the Word; and in death, although separated from each other, both remained with one and the same person of the Word.

De fide orth. 3, 27: quoted in the Catechism of the Catholic Church 626

While the Word was pre-existent, the soul of the Incarnate Word came into existence with the Incarnation, just as it does with any person: Christ was fully human.

See also this question.

I don’t understand your reference to Genesis 18. While it’s true that the three visitors are generally accepted as an allegory for the Trinity, there is no reason that one should be the Word Incarnate. With the exception of “behold, three men stood in front of him”, all the references are to “the LORD” in the singular, and it is the Lord who speaks. The three men Abraham met could just as easily — even more, likely — have been God, in the same form as he walked in the garden with Adam, accompanied by the two angels of Genesis 19:1.

Among the meanings of ψυχε (psuche, 5590; soul, life) is
"the seat of the feelings, desires, affections, aversions (our heart, soul etc.)"

I'm not certain what it would mean for my "(eternal) soul"/"living being"/"union of body and spirit" to be grieved. I think it makes much more sense to read his statement in a poetic sense, as:

"My heart is deeply grieved, to the point of death,"

Consider Genesis 6:6 (NASB)

The Lord was sorry that He had made man on the earth, and He was grieved in His heart.

Note: this is not the Hebrew word for "soul" (לֵב), only a use of the word "grieved" in reference to God.