Classic American Southern is different from working class "Southern" (which isn't actually confined to the South) more accurately called "Country" or "Redneck". It is dying out in most places in the South, although there are a few places where you can still hear it.
A few distinguishing features of Classic Southern is drawing out the end of syllables (Southern Drawl) and non-rhoticity - "r" is more like "h", and swallowing the "g" in words that end in "ng".
In a pure Southern accent, "car" is pronounced "cahhh" drawing out the non-rhotic "r"
"Water" is "wahhtuh"
"Hunting" is "Huhnt'n"
Forrest Gump had an approximation of a Southern Drawl, although not completely authentic.
In the South, classic Southern was once the equivalent of Received Pronunciation in England - spoken by the educated middle to upper class - both black and white, and has many similarities (not a coincidence). Not a working class accent. Some of the loveliest Classic Southern Accents I've heard in recent years were spoken by African Americans.
Me, I'm a half-breed: Redneck on my father's side and Classic Southern on my mother's side, so I drop half my R's and G's. LOL
The best example I can find of a lovely classic Southern accent: Bobbie Gentry. BBC gave her a TV show in the UK! Most episodes are lost.