Hi there, I think the other posters have guided you in the right direction already but I will add my two cents.
Charlotte - 2 days (Renaissance, Hornet's Nest, Winthrop, and your choice for the others)
International Disc Golf Center - 1 day
From there I would just stay along I-20 all the way to Texas. No real reason to wander too far. These are the courses that I would suggest stopping off to play along the route:
Atlanta, GA - J.P. Moseley and/or Perkerson
Birmingham, AL - George Ward and/or Iverness
Jackson, MS - The Rez
North Louisiana - Lake Claiborne State Park, Lake Bistineau State Park, and/or Ford Park in Shreveport, LA
I would allot 3 days for that part of the trip. In Texas, I would allocate your days like this:
Selah Ranch - 1 day
Trey Texas Ranch - 1 day
Dallas - 3 days
Austin - 3 days
Overall, that is 14 days with just over 20 hours of driving and you could probably get in close to 40 courses if you really wanted. If you need to shave a day or two, I would just make the drive between Augusta and Texas quicker with less stopping for disc golf in between. Texas, the IDGC, and Charlotte are the highlights of this trip. While Arkansas and Tennessee have some pretty good courses, they aren't much different from those in Mississippi and Louisiana, and definitely not worth the significant extra driving time that it would take to zig zag all over the south between Georgia and Texas. Definitely just stick to I-20 from Augusta to Texas.
That would be a killer trip!