Drk hit on a lot, but this is in my wheelhouse as the 4 states I've lived in are NC,TN,GA, and NV. I've been to harrahs Cherokee several times as well. One important question is what time of year are you planning to be there? I know you said not summer but big difference between winter,spring, and fall.

Off the top of my head.

Asheville- definitely the Biltmore house. That's a must see.
Asheville has tons of good breweries but I know that's not your thing.

Atlanta- Thomas mentioned stone mountain, it's decent, but best thing to do there is to hike to the top. Great view of the area and it's a pretty easy hike.

Atlanta - aquarium is largest in country and 2nd largest in world. Weekends are insanely crowded though so if interested try to go on a weekday. I know you said not trying to do activities for kids but I'm sure Benjamin would love it.

Atlanta - World of coke - Given your love of soft drinks this is a natural haha. It's also real close to aquarium and you can buy tickets that are good for both. The popular thing for tourists of Atlanta to do is aquarium/world of coke/cnn center all in a day as they are all together. The college football hall of fame is also new and in the are and a fun thing to do.

Atlanta has tons of good restaurants too and can give recommendations if interested. I know Thomas hates on Atlanta but I like it. Ponce City market is also something to check out as are The Beltline, and Piedmont park.

Nashville - really underrated city. I'm not a country music fan at all but the music scene there is incredible. Great food and live music at almost all restaurants. Fun city.

Memphis is pretty dang far from Cherokee, that's like a 8 hour drive with not much to do in between except Nashville.

Chattanooga has a cool downtown area on the river, but otherwise not a lot to offer.

SC - sucks as a state, other than Charleston which is awesome. I know you're not looking to go to the beach, but Charleston is a fun town on the coast, as is Savannah in GA.

NC - blue ridge parkway is a scenic highway that's a fun and pretty drive especially in the fall. Can take it from Asheville all the way up to blowing rock and Boone which are neat small towns in the mountains, with lots of sites along the way. There are also tons of different hikes you can do if you're in to that. And white water rafting/cayaking etc.

Definitely get some NC bbq.

There are tons of outdoor activities to do in that part of the country depending on the time of year. Let me know if you have any questions on anything.