Hello SandT,
It's not clear what your specific plans have worked out to be: You've noted Asheville then Bryson City as western NC destinations, not that they're far from one another or mutually exclusive in terms of seeing each in a single trip. Anyway, in keeping with your literary, civil rights, small town, outdoorsy interests, I offer the following:
Just southwest of Asheville, along the Blue Ridge Parkway, lies a viewpoint for Cold Mountain, the inspiration for Charles Frazier's novel. Access is available on foot, but it's reportedly an arduous hike.
At and below Asheville, segments of the French Broad River are paddled via rafts, kayaks, and canoes. A very nice and somewhat thrilling segment takes out in Hot Springs, NC following put-in between Asheville and Hot Springs. There is in fact a hot spring in Hot Springs, and its waters are piped into a complex of private outdoor jacuzzis which are rented out by the hour. To the best of my knowledge, Hot Springs, NC is the location of the only hot spring in the entire Southeastern and Southern part of the US. Yes, there are warm springs in GA and VA, and perhaps other locales, but for truly hot springs, it's western NC.
A wonderful paddle can be had on the Nantahala River with the take-out right at the Nantahala Outdoor Center world headquarters at Wesser, NC. I recommend the inflatable kayaks, called duckies or funyaks. The Nantahala is a great place for an inflatable kayak.
Traveling from Bryson City to Charleston, I'd stick to the main roads, including I-26, until getting out of the mountains which will be more or less at the NC-SC state line. Back road travel in the mountains can be very slow and rather frustrating. Once clear of the hills, a look at an SC highway map will reveal any number of paved state and Federal highways more or less paralleling I-26.
For a nice small-town stopover on that day's trip, consider Newberry, SC. It's far enough from Columbia to retain a largely rural character, it's the county seat, and there is a small college, Newberry College, right there in town. With a population of 10,000 or so, there should be a variety of shops and diners catering to the courthouse and general lunchtime customers. I passed through there long, long ago, and I recall it was a fairly nice place.
Between Savannah and Panama City, there are any number of small GA towns you can consider stopping at. I would certainly prefer a cross-country route to the Interstate route to Jacksonville, then across FL on I-10. To the best of my knowledge, the peach growing part of GA is further upstate, but I can't say with certainty.
Your trip dates in September are in the heart of college football season. You may be interested in checking the schedules for some of the smaller colleges and universities you'll be passing by on Saturdays. Going to a small-college football game would represent a slice of Americana difficult to duplicate elsewhere. Of particular note would be South Carolina State University, a historically black university in Orangeburg, right on your way to Charleston. The competition is sure to be spirited and the marching band is over the top.
Have fun planning and taking your Southern-fried RoadTrip!
Foy