I learned that during the American Revolution, the American accent wasn't around yet since American culture was so young at the time and that also the British accent hadn't existed yet either. If this is true, do we know what George Washington would have sounded like?
You might be interested in this answer to a similar question I posted in this sub before.
Specific to your question, yes, George Washington did have an American accent separate from what was being spoken in Great Britain at that time. But it's also true that the accents in Britain and the accents in America back then were both different from what they are today.
We are able to figure out what those accents may have sounded like by piecing it together through direct mentions in written records (which was rare to go into any detail beyond that accents existed) or, often more helpful, by reading through the phonetic spelling of half-literate people from back then.
In the former category, the first direct mention of an American accent was by Hugh Jones, an Englishman and a visiting professor to the College of William and Mary in 1724. One of the details he mentioned was that Americans had formed the habit of turning the "s" sound into "z"s in words such as raspberry and suppose, while in England they were still pronounced with a hard "s".
In the latter category, it was quite common to find phonetic spellers in America to misspell words like "world" or "force" with an "a", and see them spelled as "warld" or "farce" which gives insight into how accents sounded to people writing these words down.
We can't say for certain exactly how George Washington or the founding fathers sounded like, but through the literature we have a good idea. If you go back to that link above, I posted some links to sound examples of President William Howard Taft and New York City Mayor John F. Hylan. They were both born before 1870 and both had remnants of what in the 19th Century became termed as the "American brogue". While their accents are largely familiar, they are perceptibly different than anything heard natively in the U.S. today. So you can hear how much has changed in the 140+ years since they were born.
George Washington was born 120+ years before they were, so you can imagine how much change there was in that period, and how much it must have drifted from the Revolution until Taft was born in Ohio (with Massachusetts ancestry) about 80 years later.
In my own opinion, in fictional accounts, I would say that the best depiction of American accents of that time is the one done by Tom Wilkinson in the role of Benjamin Franklin in the HBO John Adams miniseries. It's certainly different from today's American accents in the same way that the historical literature seems to point. But of course it's a fictional account so some artistic license has been taken.
If the subject interests you, there are many sources in the link at the top of this post. The leading scholar on this subject was a man named Allen Walker Read whose essays were collected in a book called "Milestones of English in America". It's dry since the essays were originally written for scholarly journals but it is indispensable because it collects so many of the primary sources about contemporary writing about American accents from the founding of Jamestown and Plymouth until the early 1800s.
While you're waiting on other answers, you might be interested in this response, by /u/buttpoo69 (!)