TVtropes.org has articles on "Standard American English" "American Accents" as well as "Standard British English" and "British Accents" as well as an article titled "Seperated by a Common Language" which has examples of American words and phrase that may mean different things in British English, as well as the reverse. Additionally, Americans are becoming increasingly aware of British phrases and their meaning, so there articles about some "British" phrases that are exclusively about a word or phrase (Spot of Tea explains British Tea drinking culture. There's also an article on the use of "bloody" between British (where it's edgy that 11-year old Ron Weasley said "Bloody Hell") and American use (the same scenario, it's offensive that Ron said "Hell" and properly British that he qualified it as "Bloody".
Accents are especially confusing because there's a lot of difference between an A Tidewater accent, an Appalachian accent, a Southern Draw/Twang, and a Texas accent... but they all sound very similar and are all parts of the "Southern Accent" that is asocciated with former Confederate States (though Tidewater is more associtated with a union state, but good luck with that) and there are certain turns of phrases that will be indicative of that.
And yes, Americans can tell when a British person is doing a bad American accent (and you guys really can't do a respectable southern accent of any type). There's enough BBC media (Mostly Sherlock Holmes and Doctor Who... there one of the few series that got traction in the states) that we've been exposed to Brits faking it as Americans... and it's about as convincing as Americans pretending to be British. We know your mocking us... but the jokes on you cause no one talks like that. You still haven't paid us back for Dick van Dyke in Mary Poppins, though.
One thing that most Europeans really don't get is that when an American says "Have a nice day," they aren't implying anything other than exactly that they want the rest of your day to be pleasant (for some reason Europeans find this suspiciously insulting). American's as a rule will not use "good-bye" unless they're likely not expecting a return (to a point in almost any American Media that when one character says a "See ya around" or "Keep in touch" only for a response from the second that is "Good Bye," that something more is being said (either an acknowledgement that the danger of what they're gonna due means they won't make it out alive or a more insulting sneer that the visit was so unpleasant they won't return of their own free will.).
Without seeing examples of "too British" I'd can't help with pushing too say it like an American. I will say a subtle thing I do when I write British person is to use British spelling in their dialog, in contrast from American spelling in my American characters and narrative voice. Usually it's to denote a gentle and mild British accent as opposed to pidgen spelling for Cockney, Irish, or Scottish and keyboard mashing character strings of 33+ letters for Welsh (joking. I have no ear for Welsh accents and don't try them at all.).