How do Jehovah's Witnesses explain John 8:58-59 in a way that accords with their doctrine?

I think you missed the point of why they wanted to stone. Here is the reason:-

NWT John 8:57 "Then the Jews said to him: “You are not yet 50 years old, and still you have seen Abraham?”

The Jews wanted to stone him because he said he has seen "Abraham" (a claim any of the Angels can also make) which he confirmed by says he had existed before Abraham in vs 58 which has noting to do with being God as the following shows:-

FURTHER RENDERING OF JOHN 8:58 BY VARIOUS SCHOLARS AND BIBLE TRANSLATIONS

The Living Bible: “The absolute truth is that I was in existence before Abraham was ever born!”

The New Testament in the Language of Today, 1964 ed., William F. Beck: “I was in existence before Abraham was ever born”

New Believers Bible, New Living Translation, (1996): “Jesus answered, “The truth is, I existed before Abraham was even born!”

The Concise Gospel and The Acts, C.J. Christianson, (1973): “I existed even before Abraham was born.”

New Simplified Bible: “Jesus said, I tell you the truth, I existed before Abraham was born.”

An American Translation, Smith and Goodspeed, (1939): “Jesus said to them, “I tell you, I existed before Abraham was born!”

The New Testament in the Language of the People, Charles B. Williams, (1937): “Then Jesus said to them, “I most solemnly say to you, I existed before Abraham was born.”

Cotton Patch Version, (1970): “To this Jesus replied, I existed before Abraham was born.”

New Testament, Noli, M.F.S., (1961): “Jesus answered them: Well, well, I tell you, I existed before Abraham was born.”

The Original New Testament, H.J. Schonfield, (1956): “I tell you for a positive fact, I existed before Abraham was born.”

The New Testament of Our Lord and Savior Jesus Christ, George Swann, (1947): “Jesus said to them, verily, verily I say unto you, I existed before Abraham was born.”

The Clarified New Testament, P.G. Parker: “Jesus answered, before Abraham existed, I existed.”

A Translators Handbook to the Gospel of John, Nida: “Before Abraham existed, I existed, or I have existed.”

The Documents of the New Testament, G.W. Wade, (1934): “Jesus said to them, in very truth I tell you, before Abraham came into being, I have existed.”

The Complete Gospels Annotated Scholars Version, Miller, (1992): “I existed before there was an Abraham.”

The Bible, A New Translation, Dr. James Moffatt, (1935): “Truly, truly I tell you,” said Jesus, “I have existed before Abraham was born.”

The New Testament Or Rather the New Covenant, Sharpe, (1881): “I was before Abraham was born.”

The Worldwide English New Testament Bible: “Jesus answered, I tell you the truth. I already was before Abraham was born.”

Good News for the World, (1969): “Jesus answered, I tell you the truth. I already was before Abraham was born.”

International English Version, (2001): “I was alive before Abraham was born.”

International Bible Translators, (1981): “Jesus said to them, I am telling the truth: I was alive before Abraham was born!”

The Simple English Bible, (1978): “Jesus said to them, I tell you the truth: I was alive before Abraham was born.”

The Four Gospels and Revelation, Richmond Lattimore, (1979): “Truly, truly I tell you, I am from before Abraham was born.”

New Covenant, J.W. Hanson, (1884): “Jesus said to them, truly, truly, I say to you, I am before Abraham was born.”

A Literal Translation from the Syriac Peshito Version, James Murdock, D.D., from 5th century manuscripts, (1896): “Jesus said to them: Verily, verily I say to you, That before Abraham existed, I was.”

Twentieth Century New Testament: “In truth I tell you,” replied Jesus, “before Abraham existed I was.”

The New Testament According To The Eastern Text, George Lamsa Translation, (1940): “Jesus said to them, Truly, truly, I say to you, Before Abraham was born, I was.”

The Curetonian Version of the Four Gospels, Burkitt, from 5th century manuscripts, (1904): “Before Abraham came to be, I was.”

The Old Georgian Version of the Gospel of John, P. Blake, M. Briere, in Patrologia Orientallis, Vol. XXVI, faxcicle 4, Paris, from 5th century manuscripts, (1950): “Before Abraham came to be, I was.”

Ethiopic-Edition: Nouvum Testamentum Æthiopice, T.P. Platt, revised by F. Praetorius, Lepzig, (1899: “Before Abraham was born, I was”

The New Testament, Curt Stage, (1907): “Before Abraham came to be, I was.”

The New Testament, Kleist & Lilly, (1956): “I tell you the plain truth. replied Jesus, I am here - and I was before Abraham.”

New American Standard Bible, 1963 and 1971 editions alternative rendering: “I have been”

The New Testament, George Rh. Noyes, D.D., “Professor Of Hebrew and Other Oriental Languages and Dexter Lecturer on Biblical Literature in Harvard University”, (1869): “Jesus said, truly I tell you, from before Abraham was, I have been.”

“The Four Gospels” According to the Sinaitic Palimpsest, Agnes Smith Lewis, from a 4th-5th century manuscript, (1886): “He said unto them, 'Verily, verily, I say unto you, Before Abraham was, I have been.”

The New Testament in Hebrew, Franz Delitzsch, 1937 edition: “Before Abraham was, I have been.”

The Unvarnished New Testament, Andy Gaus: “Truly, truly I tell you, Before Abraham was born, I have already been.”

The New Testament (in German), Friedreich Pfaefflin, (1949): “Jesus: Before there was an Abraham, I was already there.”

The New Testament in Hebrew, Isaac Salkinson and David Ginsberg, 1941 edition: “I have been when there had as yet been no Abraham.”

Translation of New Testament, Wakefield, G., (1795): “Jesus said unto them: Verily verily I say unto you, before Abraham was born, I am He.”

Ledyard, G.H. New Life Testament, (1969): “Jesus said to them, for sure I tell you, before Abraham was born, I was and sum and always will be.”

The Coptic Version of the New Testament in the Southern Dialect, George William Horner, (1911): “Before Abraham became, I, I, am being.”

The New Testament, James A. Klist, S.J., and Joseph L. Lilly, C.M., (1954): “I am here - and I was before Abraham.”

“Jesus said to them, Truly, truly do I say to you, From before Abraham was, I have been.”-THE NEW TESMAMENT TRANSLATED FROM THE GREEK TEXT OF TESCHENDORF BY GEORGE R. NOYES, D.D., HANCOCK PROFESSOR OF HEBREW AND OTHER ORIENTAL LANGUAGES, AND DEXTER LECTURER ON BIBLICAL LITERATURE, IN HARVARD UNIVERSITY.

I actually do not see a problem here. The Jehovah's Witness group teaches that God the Father created Jesus first (source), then Jesus created everything else (source). So they do believe that Jesus existed before his incarnation.

To the Jews, this is an irrelevant distinction. It is blasphemous either way.

1. "Jesus is God" = blasphemy - stone him
2. "Jesus was created by God before everything else" = blasphemy - stone him.

Anything about "I am" being in reference to God the Father, explicitly, is simply denied and translated accordingly in their New World Translation. Whether their translation is valid or not is a question for the Hermeneutics site.