You need altitude to get cool. Mt Pisgah NPS campground on the Blue Ridge Parkway is over a mile high, and is very comfortable in the summer. No hook ups however, just a dump station, a few faucets, and showers and restrooms. Go to Ashville and go south on the Blue Ridge Parkway from there. You dont say what your rig is, but you do have to pass thru a couple of tunnels to get to the campground from Ashville. several waterfalls in the area you can drive to and easy walk to see them. Do not attempt to access the parkway with a motorhome or trailer via US 276 or state highway 215, both of which go up the side of the mountain.
List of Tunnels Along the Blue Ridge Parkway
Another choice is Georgia's highest state park. Black Rock Mtn, located at Mountain City on US 23 south of Franklin, NC. The campground is 3400 ft and has a steep 8% or so grade for a mile and a half or so going up to it. There are some long spaces, some as deep as 60 ft, however the reservations system lists them all as 25 ft. Good showers and restrooms and walking trails, scenic views, and several local produce stands to get really fresh veggies and stuff. The Foxfire museum and heritage center is located in Mountain City. https://www.foxfire.org/visit-us/