I have warm memories of cooking up grilled cheese in an iron skillet with lots of butter and sourdough bread and Tillamook cheddar. Ah, the years of living in the wilderness!
Tillamook cheddar is certainly delicious, but it comes from up here in the PNW, in Oregon. I doubt that it's traditionally Southern, and neither is sourdough, although cooking in an iron skillet is traditional.
As far as "authenticity," I don't know of any good cheddar cheeses from Georgia, but a nice sharp cheddar that could be used in an old recipe may have come from Vermont. Other than that, I'd go the Paula Deen route and use Velveeta, which many, many Southern cooks used for their Pimento Cheese, or "Menna Cheese." Tabasco, pickles, pimiento peppers and mayonnaise were added to make up for the flavor that Velveeta lacks.
Other than a grilled pimento cheese sandwich, as I've described above, I'm unaware of any distinctly Southern-style grilled cheese. So go grab yourself a tub of the stuff at the store, or get a log of Velveeta and mix yourself up a vat of Paula Deen happiness!
This isn't Paula's personal recipe, because she's kind of cleaned up her act lately, but it's close - http://www.southernplate.com/2009/10/grandma-lucys-pimento-cheese-its-a-southern-thang.html