We live in Indiana... about half way between Ashville, NC and Sturges, SD, and we have been to both. Actually, I lived in Morganton, NC for a few years about 35 years ago. We travel your half of the route to South Carolina and/or Florida twice a year.
Last Summer we went to Mount Rushmore from Indiana. Pretty much the same route you've selected.
Because we were traveling, we took 3 nights to get to Custer, SD and 4th night we were at our destination. Going South, we take 2 days (1 over night) to reach South Carolina or Florida (depending).
For overnights, anywhere along those routes, ANY KOA campground will accomidate you for an over night. We found the closer we got to South Dakota, the more other private campground required a 2 or 3 night stay.
Between a combination of State Parks and KOA's for an overnight, you can take any of your pick and you won't go wrong. FYI, we planned on actual travel time, only 200 miles a day for the South Dakota trip. That way, we could unhitch the trailer at the KOA and still have most of the afternoon and all evening to explore the area and do things. Then we still had over a week at Custer, Mount Rushmore, Sturges, the Bad Lands, and traveling the Black Hills, Deadwood, all those places. It was great!
So, my suggestion, pick the distance you want to travel per day, and look for a KOA in the area and do an over night. All KOA's will let you do a one night over night. In - out - there - gone! Full hookup, and affordable.
Good luck.
