US 441 from Gatlinburg, TN to Cherokee, NC for any practical purposes has no shoulders.
AFAIK, none of the routes that cross the Appalachian crest between TN and NC have a shoulder, except I-26. Not on US 58, 421, 321, 19E, 25/70, 441, 129, or 64; not on Cherohala (165/143) 32, 107/226, 143/261; and not on I-40 (not that you can bicycle that anyway)
(1) I know a couple people who have cycled the Newfound Gap road (441). Traffic is generally not too bad early (up 'til 10:00 or so), and half the road is downhill, where few drivers will even attempt to overtake.
(2) Cherohala and U.S. 129 are lightly trafficked except on weekends, as is 25/70 through Newport. Most of the traffic through Newport takes I-40, unless there's a wreck, landslide, or road construction (i.e., every other year).
(3) Also, the road from Roan Mountain up to Roan Mountain and down to Burnsville (143/261) sees mostly dedicated sight-seers headed for Roan High Knob on the twisting climb up the Tennessee side, so it's more or less tolerable. Make sure your brakes are in good shape before you start the descent into NC.
(4) The section of BR 21 going south out of Chattanooga is one I've ridden several parts of, although I'd suggest taking Tennessee Ave. off St. Elmo to the state line. Even though St. Elmo Ave. has striped bike lanes, it's a two-lane drag strip, and the bike lanes go through storm drains most of the way. Broad St. is OK except at rush hour.