The early Americans didn't change their accents so they would no longer sound British, the British changed their accents on purpose because the Industrial Revolution back home was making commoners rich, and a rich commoner could buy anything a rich aristocrat could. The British gentry reckoned that if they no longer could look different from the commoners, they at least could still sound different, so they premeditatedly changed their accent as a shibboleth, a sort of linguistic Masonic handshake.
The closest accent to that a British gentleman would have used in the late 18th Century is what is spoken today by the elder members of any First Family of Virginia.
"The closest accent to that a British gentleman would have used in the late 18th Century is what is spoken today by the elder members of any First Family of Virginia."
Reminds me of what Val Kilmer said about his portrayal of Doc Holliday in the movie 'Tombstone.' He said Holliday's dialect is believed to have been that of a Georgia aristocrat. Although no one uses that dialect today, some of Kilmer's older relatives remembered older people speaking that way.