Check these out on your atlas - -

Starting at Fort McAllister State Park

Georgia -
GA 144, GA 178, GA 147, CO 95, GA 117, US 280, GA 118, US 82, GA 266

Alabama -
AL 10, AL 93, US 29, (or country roads from Brundidge thru Spring Hill to Luverne)
AL 10, AL 28, AL 25, AL 14

Mississippi -
MS 69, MS 50, MS 9, MS 404, US 51, MS 8, MS 1, US 49

Arkansas -
US 49, AR1, AR 261, AR 78, AR 306, AR 17, AR 14, AR 125

Missouri -
MO 125, Bike 76 to border, MO 126

Kansas -
KS 126, Bike 76 to Neal, US 54, KS 99, KS 18

For free camping in the deep south - ask folks in town or at the crossroads store where a fishing access site is. It will be a levee or bayou - there will be mosquitos - and you may have bubba company - but it's free.

If you want to camp in state parks you can do some nearby routes, but I'm guessing you will be stealth camping most of the way.

Most of these roads have moderate to low to very low AADTs.