If you have the time, US-70 aka Broadway of America would be a great choice. Since it runs parallel to I-40 you have the option to jump on if the pace gets too slow. I would recommend getting off 70 in Lenoir City and take 321 to Maryville, the 411 to 441 which takes you into Sevierville. This eliminates Knoxville completely.
We recently left the area via Cosby and the Foothills Parkway, stopping at the Cosby part of the Nt'l Park and the Carver's Apple House. Really enjoyed our time. It, too, was a lovely drive. Took longer to get home, but liked the variety. We plan to stop and eat near Douglas Lake in the future.
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