>There is no reason to think that the stories were not written down immediately near the events. After all, these were literate people. The archeology of the regions reveals all kinds of writings, methods, transcriptions, scribes, etc.
Look to your earliest Garden of Eden stories. There was no writing communication for thousands of years after these events.
Elementary pictographic "writing" by a few professional scribes by the time of the Exodus, certainly, but much earlier than that, no.
Are you accepting a biblical dating or are you accepting standard archeological dating?
Sumerian writing goes way back.
The issue of Noah and earlier is, of course, changed by whether or not we accept a cataclysmic universal flood. If so, then there are no records...even if they kept records.
If you accept a biblical dating scheme, then Shem lived into the Abrahamic times. First hand accounts could be written from an observer.