I've been reading a good book on this, let me summarize what I've learned and I'll use quotes as well:

In the late 1700s American English was just as different from today's as it was different from British English. "Though it would be clearly identifiable as English, it would be a variety of English unlike any we had heard before."

Words with a "kn" like knee would be pronounced like "t'nee" (only a generation before the founding fathers, Americans pronounced "kuh-nee")

The "ah" sound in words like father was not present. Instead, father would rhyme with modern lather.

Was was still being pronounced like "wass", but Kiss sounded like modern "Is" with the z sound at the end.

"Speech was in general much broader, with more emphatic stresses and a greater rounding of r's. ... Interior vowels and consonants were more frequently suppressed, so that nimbly because 'nimly' and somewhat was 'summat'."

You definitely wouldn't hear anyone say "thee" or "thou". That practice had been outdated for almost 100 years, replaced by the more polite "you."

I could go on and on about the pronunciation differences. The point is that they definitely did not sound British. My educated guess (I'm not a linguist or historian) is that it sounded more like the modern Boston/New England accent than anything else.

All this being said, you should know that, back then, the states were much more separated than they are today. The accent in Virginia would be very different from the accent in the Carolinas. Also, the speeches you hear today are taken from written text by these men. They wrote differently than how they spoke, and your reenactments are most likely just trying to read the text how it was written, not how it would have been spoken.