How did we get this far without even mentioning Charleston. From a pure disc golf course perspective, there are better cities. But Charleston is good enough disc-wise. Considering all, would you want to live in Augusta over Charleston? Not in a million years.
Other options would include Austin or Tampa/Clearwater, if a slightly bigger city is ok.
Nashville, Charlotte, etc, are great for year round living. But specifically as winter getway.........I'd go much further south.
O.P.'s criteria included "in the heart of a bunch of great disc golf courses." While Charleston's one of my favorite places in the world, and has a very good course and several decent-to-good courses, I assume he's looking for more than that, disc-golf-wise.