I traveled Chattanooga to Asheville - with a jog over to Hiawassee, Georgia for the Springstream Rally - this past March. This was along 74 and some other such non-interstate roadways.
The ride over the mountains Georgia to Asheville was exhilarating, to say the least. My rig is heavily loaded and I was slowed down to 15mph on one climb. Some of those long 10% grades down are white knuckle exhilarating too.
They probably do a better job of smoothing out mountain grades on the interstates but things are more interesting the back country way.
If you have the time take the scenic route.
I was on the road for 173 days and your state and these parts of Tennessee and North Carolina are about as beautiful as anything your country has to offer.
Here's another recent thread on Asheville. Note my post at #12:
Mama Gertie's, notwithstanding the name, was the best park I stayed at during that 6 month southern travel.
Asheville is a very cool place. Enjoy.