Best route from Murphy, NC to Knoxsville, TN

Looking at my Rand Mc Nally,I see one route. US 74 to I 75.

With a large RV, I'd avoid US74 WEST to I75, which will run you through the Ocoee River Valley. Pretty drive, but if you want to avoid tight turns it's not the route I'd pick. While it would be around an hour longer, I'd consider taking US74 EAST from Murphy, through Bryson City and picking up I-40 to head west to Knoxville. Other than a fairly short stretch through the Nantahala, it's primarily 4 laned and not very twisty.

74/64 east from Murphy to I40. You'll go through the Nantahala gouge but it's not a bad drive. For the most part it's four lane. I think it's exit 29 on to I40.

Both the 74 east and 74 west will have you on single lane mtn roads for about 21 miles. 74 west however will have you on single lane road for about an additional 5 miles before you get back to dual lane roads. 74 west will be a harder drive since 74 east is follows mainly along the river so not nearly as much up/down. 74 east will mean a few more steep grades up/down between Franklin and Sylva. Even when you get to I40, you will still need to drive through the Pigeon River Gorge which adds some twist. Under normal circumstances, I would say the routes are about even with each other. However, I40 is still down to one lane in both directions until about May 1. This to me would make me choose 74 west and deal with 20 miles of more difficult road but then it would be clear sailing up I75. I40 could be a clear shot through or you could have 2 hour delay because of the reduced lane.

My choice would be TN-68 from Ducktown TN to Sweetwater TN. The part of 68 from Ducktown to Telico Plains will be crooked, but no worse than US-74 thru either Ocoee or Nantahala. From Telico Plains to I-75 is wide and pretty straight. The only way out of Murphy that doesn't have crooked road is GA-515 south to Atlanta.

Had an old guy years ago when I was a kid tell me, "son, I don't think you can get there from here".
Basically, you are going to find a few curves any way you go. The 68 route is the best up to I-75. That 74 route is pretty in an auto but I would tow or take my MH across there.

You could go south on 19 to Blairsville, then west to Hiawassee and then to Clayton and north on 441 to Franklin NC to Silva and Waynesville and go I-40 west thru the gorge up to Knoxville. That would be a huge amount out of the way. No switchbacks but still mountain roads for the most part.