There are a couple small museums in K'ville and the Downtown area is very walkable and lots of good restaurants. A museum outside of Townsend which is near Cades and a cave but not sure if its open. Cherokee has the museum of the Cherokee Indian and Waynesville has wheels through time museum which is great. Biltmore is a wonderful place but pricey, still worth it to see once in your life. You could do a day trip up the Blue Ridge Parkway to Mt. Mitchell, the highest point in NC and its only a little north of Asheville. It has a restaurant for a good lunch there too.
Bryson City has a train you can ride on through the mountains. There is the museum of energy in Oak Ridge. Lots of antique stores in Clinton. There is Marble Springs, the home of TN first governor John Sevier. There is a new section of the Foothills Parkway open between Walland and Wears Valley with spectacular views. Then there is always the shows to go see in PF and G'burg. Sweet Fannie Adams is a nice little Vaudville style show and there is the aquarium too. Lots to do if you seek it out.