What was Jesus' soul? In the following Scriptures Jesus refers to my soul:

Matthew 26:38  KJV


  Then saith he unto them, My soul is exceeding sorrowful, even unto death: tarry ye here, and watch with me.


Mark 14:34  KJV


  And saith unto them, My soul is exceeding sorrowful unto death: tarry ye here, and watch. 


John 12:27  KJV


  Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour. 


In Genesis chapter 2 we are told that man became a living soul when God breathed the breath of life into his nostrils.

Genesis 2:7  KJV


  And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul. 


Succeeding generations apparently become living souls when their nostrils receive the breath of life.

In Genesis chapter 18 it appears that a pre-incarnate Jesus spoke with Abraham, since no Angel would be able to modify orders for the destruction of Sodom.

Genesis 18:26  KJV


  And the LORD said, If I find in Sodom fifty righteous within the city, then I will spare all the place for their sakes.


Since Jesus the deity was there before creation Jesus the man certainly was not referring to that as his soul.

So I have wondered if a soul was formed for the human part of Jesus when he was born of Mary, or is there another Scripture I have overlooked, or some writings on this that I do not know about.
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.