I’d dump Gatlinburg completely - it always struck me as a huge tourist trap. And the Smokey Mountain National Park used to be the most visited (crowded) in the country. Move a day or two to Chattanooga - Lookout Mountain (ride up the cable railroad). Depending on your interests, the National Battlefield Park, the railroad museum, the river; perhaps another day to Knoxville. That would save you a bunch of driving. (Tennessee is LOOOOONG). The Holston valley is attractive, but you’ll have plenty of mountain driving without it.
Backing up a bit. Memphis - mud island used to be nice, but my visit there was so far back they still had the Memphis Belle there. I don’t know if it is still worth the trip.
Nashville - the Hermitage, if you are into history. And/or visit the boyhood home of the man who won the War Between the States - Elm Springs, in Columbia, Tennessee, less than an hour south of Nashville (open weekdays only)
Chattanooga - if you want nature with out the “Hillbilly Golf” attractions, there’s Cloudland Canyon State Park in Georgia up on top of Lookout Mountain. http://gastateparks.org/CloudlandCanyon
See Rock City at Lookout Mountain. Chattanooga. Way cool for the kids.