True in the overall sense, but the final scene was purely for dramatic effect. The vote took place on July 2, 1776, and thereafter the idea was hatched to put it in writing, and the writing itself was not signed, according to the records, until August 2, 1776.
Adams expected everyone to be celebrating the event on July 2, 1776 every year. This is an inaccuracy irrelevant to the story, though, IMHO.
I wonder if there is an intrepid sort that could make an exciting modern (current day) version of this story, so that when the kiddies at school are looking for a lazy history lesson, they can show the movie and take a “test” after watching it...you know, so the socially promoted ones who can’t read can learn it, too.
It is an important story.
Will never happen. Hollywood and the NEA have successfully promulgated the meme that the Founding Fathers were misogynistic racists and backwards womyn haters.