Are Angels more accessible than Jesus?

Scripture doesn't teach us to pray to angels nor even to Jesus, nor to the Holy Spirit, but to the Father alone. Jesus himself taught us to pray to the Father. "After this manner therefore pray ye: Our Father which art in heaven..." (Matthew 6:9) He also taught us to pray to the Father in his name (John 14:13-14). This is why Protestants begin their prayers addressed to the Father and end with "in Jesus' name," or "in Christ's name. Amen." (Aside, perhaps, from some Methodists, Anglicans, etc. that retain Catholic customs without any scriptural proof that these are valid.)

Scripture does seem to teach that angels carry our prayers to God (Revelation 5:8, 8:3-4) but this doesn't mean we should pray to them. We pray to God alone, even if angels are somehow involved in relaying the message (although, its probably just metaphor, especially considering its in the highly figurative Book of Revelation).

Are Angels more accessible than Jesus?

A. Not after the coming of Our LORD Jesus Christ.

Heb 8:11 (RSVCE)
11 And they shall not teach every one his fellow or every one his brother, saying, ‘Know the Lord,’ for all shall know me, from the least of them to the greatest.^[a]

Footnotes:
^{a. 8.11 This verse means merely that knowledge of God will be commonly shared. It does not exclude the existence of a ministry of teaching in the Messianic times nor negate the communion of saints (my addition and emphasis) - cf. CCC Credo Chart.}

God and his angel and his saints, and God in his angels and in his saints, are family. We still get to talk, commune with one another (just as parts of the body do) even when we have access to the Father through Jesus Christ¹. God wants and likes it that way too; to listen and grant us favors through his friends, especially the angels and saints - the cloud of witnesses, those who have gone before us [marked] with the sign of faith, those who have made it - in heaven, thereby glorifying them.

^{1. Catholics can talk with him whenever they receive him (even daily if possible) in the Eucharist or whenever they visit him daily or as often they can or want to in the Blessed Sacrament of the Altar reserved in the Tabernacle.}

There is a morning offering that concludes:

[...]
May I gain all the indulgences that I can today.
Holy Mary, be a mother to me.
Angels and saints of God, pray for me.
+ May our LORD bless us and preserve us from all evil and bring us to everlasting life. Amen.
+ May the souls of the faithful departed through the mercy of God rest in peace. Amen.

For the phenomena you have experienced and continue to experience, if catholic, better consult with a priest.

1 Thessalonians 5:25 Brethren, pray for us.

One has to ask: why would apostles of Jesus Christ ask fellow Christians to pray for them, instead of simply asking Christ directly? Unless one answers this question and accepts the answer, one will never understand Catholic teaching on the "communion of the saints" (Apostle's Creed). It's not superfluous, so what is its purpose? The same principle is at play when the apostles ask their fellow Christians to pray for them, as when we ask ours.

One answer, certainly, if not the answer, is that the apostles, like any Christian, have a certain duty before God - the duty to live as Christ wants, with the graces given them by Him (cf. the Parable of the Talents). Given this, it follows that if a Christian wishes to offer their own suffering, prayer, sacrifice for another, that God will honor that by granting their fellow with the graces that would be afford them themselves (what else does "This kind cannot by expelled except by prayer and fasting" Mk 9:29 mean, or, "I rejoice in my sufferings for you, and I fill up in my flesh what is wanting with regard to the sufferings of Christ, for the sake of his body, the church?" Col 1:24). We are one body, and so can help each other:

1 Corinthians 12: 12-26 For as the body is one, and hath many members; and all the members of the body, whereas they are many, yet are one body, so also is Christ. 13 For in one Spirit were we all baptized into one body, whether Jews or Gentiles, whether bond or free; and in one Spirit we have all been made to drink. 14 For the body also is not one member, but many. 15 If the foot should say, because I am not the hand, I am not of the body; is it therefore not of the body? 16 And if the ear should say, because I am not the eye, I am not of the body; is it therefore not of the body? 17 If the whole body were the eye, where would be the hearing? If the whole were hearing, where would be the smelling? 18 But now God hath set the members every one of them in the body as it hath pleased him. 19 And if they all were one member, where would be the body? 20 But now there are many members indeed, yet one body. 21 And the eye cannot say to the hand: I need not thy help; nor again the head to the feet: I have no need of you. 22 Yea, much more those that seem to be the more feeble members of the body, are more necessary. 23 And such as we think to be the less honourable members of the body, about these we put more abundant honour; and those that are our uncomely parts, have more abundant comeliness. 24 But our comely parts have no need: but God hath tempered the body together, giving to that which wanted the more abundant honour, 25 That there might be no schism in the body; but the members might be mutually careful one for another. 26 And if one member suffer any thing, all the members suffer with it; or if one member glory, all the members rejoice with it.

The church is an 'organism,' not a set of individual, independant 'cells.' Therefore, so ask for the help of fellow Christians - even if they are of the angelic, not human, nature - is to admit and acknowledge and implement this fact in your life as a Christian.

It happens also to be the practice of the church immemorial.

There is also the aspect of our being more worthy of the audience of angels and saints than God Almighty, by default, and at all times, without exception, making recourse to angels and saints in heaven, the more humble option, when beseeching God's mercy or grace.

Worshiping the Angels is condemned by the Bible. Colossians 2:18 says "Let no one disqualify you, insisting on asceticism and worship of angels, going on in detail about visions, puffed up without reason by his sensuous mind". So we can not accuse Catholics of Angel worship. The actual term used by the Catholics for Angels and Saints is veneration (Dulia in Greek). This point is reiterated by the Catholic Church over and again. Catechism of the Catholic Church mentions that "In the meantime, the whole life of the Church benefits from the mysterious and powerful help of angels" (Part One :The Profession of Faith, Section Two, Chapter One,Article I,Paragraph 5, Verse 334). So the Bible prohibits Angel worship and the Catholic Church says that it benefits from the Angels.

How can we understand this? The Angels have a mediation power. They are requesting to God for our cause but do not have the ability to interfere with the decision making process of God. The need for such a mediation arises from the fact that humans who are sinners and imperfect are not worthy of direct contact with God. We better channel our requests through a mediator who at least had a vision of God. The mediators are themselves imperfect and not at par with God in any way. But at least such an attempt looks more humble.

Why would one choose to pray to an Angel or a Guardian Angel over Jesus at times?

When Catholics say they pray to a saint or angel in heaven, that means they are asking these creatures to pray for them. It does not mean they are worshiping the creatures.

Now why ought we call upon the saints to pray for us? St. Thomas Aquinas says:

According to Dionysius (Eccl. Hier. v) the order established by God among things is that "the last should be led to God by those that are midway between." Wherefore, since the saints who are in heaven are nearest to God, the order of the Divine law requires that we, who while we remain in the body are pilgrims from the Lord, should be brought back to God by the saints who are between us and Him: and this happens when the Divine goodness pours forth its effect into us through them. And since our return to God should correspond to the outflow of His boons upon us, just as the Divine favors reach us by means of the saints intercession, so should we, by their means, be brought back to God, that we may receive His favors again. Hence it is that we make them our intercessors with God, and our mediators as it were, when we ask them to pray for us.

Yes, there is only one Mediator (1 Tim. 2:5), but that does not mean there aren't other mediators subjected to Him. Think about who brought you to Christ. You didn't do it alone. Saints and angels in heaven plus people on earth can help you through their prayers and actions. These people are your mediators, being subject to the one Mediator.

The Blessed Virgin Mother of God is Mediatrix because without her "Be it done to me according to thy word" (Luke 1:38)—her fiat—there would be no Christ, no Church, and thus no salvation. She brings us to Christ and Christ to us in a most supreme way than any other creature can.