I've been driving I-40, the same stretch, at least once every couple years since 1979. Sometimes, 3 or 4 times a year. I've towed U-hauls, a car on a car carrier, pop-up's, utility trailers, 3 different travel trailers, and lot of trip on a motorcycle, a Volkswagon, a 3500 dully, a Chevy S-10, all sorts of mini-vans, cars, and SUV's.
The road has not changed since I traveled it the first time. Nothing has changed, except maybe it's paved a little smoother now. Same curves, same mountains, same path. The only difference I've noticed over the years, the big 18 wheeler semi-truck drivers seem to much more aggressive than they use to be.
DRIVE THE SPEEDLIMIT or slower and you'll have no problems at all. If you attempt to speed, you'll find yourself in trouble going around some of those curves. They don't put truck run-away slips there for the scenic view!
If you have exhaust brake, use it. Avoid using brakes as much as possible on the down-hill side, use lower gears and just let the engine roar. As long as the engine is not red-lining it's OK.
Stay in your lane, avoid passing vehicles. Some of those curves are just not good places to pass. Most important, don't get in a rush. It's a beautiful drive any season of the year. We've got photos of almost every inch of that stretch over the years from all 4 seasons, and each time we travel through it's still a marvelous experience. You'll have 2 tunnels going East, one tunnel going west. Even the tunnels are really interesting. AND no, you don't have to turn your propane off to go through them.
If the semi-trucks can drive that stretch, and there are many, many, many that do, you'll have no problems either. Any season of the year we've seen all kinds of RV's on that road.
Enjoy and have your camera ready.
Edit:
I moved to North Carolina from Indiana in 1979 and lived there for 2 years. Lots of trips back and forth from home (Indiana) and my work in NC. Later in life, I was in the Army, stationed on the East Coast. Traveled I-40 many times going home. Later, after I got married, we traveled that road for vacations with pop-up and travel trailers. And again, even later, my daughter was in the Navy and stationed in Charleston, SC and later Norfork, VA. And yes, lots of trips back and forth when she lived there. And finally, we travel twice a year now, December and March for our annual trips the week of Christmas and the week of our wedding anniversary. We hit state parks in Florida or South Carolina.