Tom,
Memphis is over 300 miles west of Sevierville, unless you go out of your way it won't be along your route.
Last year when I went to the National Rally held in Lancaster, PA, I took an extra 5 days and traveled the Blue Ridge Parkway and Skyline Drive. It was a very enjoyable trip and at a leisurely pace. One of the nights we stayed at Big Meadows Campground and watched the deer walking around in the campground.
From the Parkway, you could take numerous side trips. Didn't see too many places with southern barbecue but we had a regional
Appalachian Campers Spring Rallyat Meadows of Dan and took a little side trip to Mt. Airy and had a great Pork Chop sandwich at Snappy Lunch, which is next door to Floyd's Barber Shop, home of Andy Griffith.
Lot's of stuff to see and do if you have the time.