The oldest extract from the Old Testament we have is in the form of two very ancient silver scrolls dating from before the Jewish exile. The scrolls accurately quote parts of the bible and were included in a burial. The second oldest evidence is the Dead Sea Scrolls from a few centuries before Christ. These show that the Old Testament in Hebrew and Greek has not changed in 2000 years.
The events in Genesis predate the existence of a written form of Hebrew, but much of it does not predate the earliest written forms of Egyptian. Moses was raised in Egypt and the Israelites spent 400 years there. It would be centuries after they settled in Israel that the Hebrews would develop their own written language.
I think the various explanations for why Moses is not the author of the first five books of the bible, despite ancient tradition saying he was, are relatively weak. If he were the author, he almost certainly wrote the books in Egyptian script, because it existed at the time, just as the New Testament was written in Greek, despite Jesus likely speaking Aramaic, due to it being a common language in the Roman Empire at the time. Should Moses have written the original five books of the bible down, then they very likely would have had to be rewritten into Hebrew hundreds of years later when Hebrew came into being as a written language.
We know that there was some form of written language among the Israelites hundreds of years before they developed Hebrew written forms, because God gave them the Ten Commandments in a written form. I would assume these commandments were written in an Egyptian written script, given the time period and the lack of a Hebrew alphabet at the time.
Another major thing often used against Moses' authorship is that the bible says that at a certain time the Canaanites occupied Canaan. This is used to say that the author knew this was no longer the case, but Moses never entered the promised land, he could only have relayed what was passed down to him. For all he knew, Canaan was differently occupied, which could explain the verse. Another good explanation would be that it was added as a later gloss, perhaps when translating his words into written Hebrew.
Regardless, if Moses did codify the first five books of the bible into a written form, or had it codified, it is almost certain he would have done so based on longstanding oral traditions for the book of Genesis. As modern research finds, oral histories can be passed down from generation to generation intact, especially among preliterate peoples. The Iliad and Odyssey codified by Homer are thought to be accounts with historic basis of the Trojan war thanks to modern archaeology, albeit fantastical stories are included. Moses or whomever compiled the accounts from Jewish prehistory, passed down over hundreds of years, would have had a relatively accurate means of gaining history via oral tradition.
The manner in which the dialects in the early Old Testament seem to show parts from as old as 950 BC and others, small segments, as new as about 500, can as is often suggested be due to oral histories being recorded in writing, or written accounts being compiled. But this should not in fact rule out Moses authoring these books, given that if he did author them it would not have been in written Hebrew but in Egyptian script which existed at the time of Moses. Given the timing of events, the Ten Commandments written by God on stone tablets very likely would have also been written in an Egyptian script. After all, the Israelites were living in Egypt for hundreds of years at this point.
We know that the Septuagint and the Hebrew Bible that we have today, both also recorded in the Dead Sea Scrolls, likely originated from some common earlier script, given that the Septuagint was translated, seemingly with the same translation by between 70 and 72 Jewish translators, into Greek for the library of Alexandria, with the translation of the various scribes being likewise, suggestive that the Septuagint was translating a Hebrew version of the bible mostly the same as the Hebrew one we have now, but with the differences between it and the Septuagint we know are there.
If Moses did, as tradition tells us, write those first five books of the bible, that would have had to have been translated into written Hebrew at a later stage, and perhaps would have been updated at each point to be understood by the people of that time, when a part of the language became too obscure. What we see in the texts we now have is exactly what one would expect to see if the original five books of the Hebrew bible were written by Moses, because written Hebrew didn't exist yet, but written Egyptian did, and Moses was literate, he understood the stone tablets of the Ten Commandments, having been raised by a member of Egypt's upper class and thus likely having had his training in written language being in Egyptian.