Anybody from the South can spot a fraud. Even people in the individual states have slightly different variations of the Southern accent. To me, TN and Texas sound a lot a like. Alabama and Georgia have their own accent.
Well, you have the coastal accent, of which Savannah is a subset, then there is Augusta, very specific local sound. The Piedmont is probably the closest to a "standard" southern accent, spoken from Columbus to Macon and points thereabouts. Rome has its own quirks, as does the northeast area up around Young Harris, Blairsville, and Mineral Bluff (that's your "mountainy accent". And Old Atlanta (not all the corporate types who are there on their way to somewhere else) has a distinct sound as well. Since I *am* old Atlanta, I can tell you that there are two "shibboleths" by which we recognize one another . . . the pronunciation of "Piedmont Road", and the pronunciation of "Ponce de Leon Avenue." You can't fake it, you take it in with your mother's milk.
There are at least four general accents in Georgia, if you don't count very local dialect survivals like Geechee and Gullah.
Joel Chandler Harris of "Uncle Remus" fame was instrumental in recording many of those accents before they vanished. Read "Daddy Jake, the Runaway".