Are there any other historical accounts that document what happened during or shortly after the tower of Babel? I was wondering if there are any other accounts from the people's that were dispersed, and how exactly they reacted to the event and built new cultures with their new languages.
There are other accounts, which may be considered to tell the same story. Wikipedia provides a good list.
Whether any of the accounts (including the Biblical one) are historical is entirely another matter. Most historians (outside of a young-earth tradition) view the myths as (primarily*) fictional, meant to explain the existence of languages to an otherwise naïve audience. Historians take this view because the history of languages is fairly well understood and known to predate architecture and human civilization by hundreds of thousands of years.
*This doesn't mean there isn't some element of truth to the stories. I.e. there may have been a tower under construction, which was imbued with additional meaning. But the view is that the story as the origin-of-human-languages is fictional.