We have visited this area twice.
Devil's Tower-I would most definitely stay at the Devil's Tower/Black Hills KOA. You are right beside Devil's Tower. You can explore Devil's Tower, and then walk back to your rv. It took my husband and me an hour or two to walk around Devil's Tower. We then relaxed at our rv and enjoyed the view. This is out in the middle of nowhere, so it's a great place to relax and do some laundry if you have downtime.
Glacier-We stayed inside the park on the east side at St. Mary's Campground. No hook-ups. There was a place to dump and get fresh water close to the campground. St. Mary's was within walking distance of the shuttle and a visitor's center (there was wi-fi at the visitor's center). I would take the shuttle and not drive. The park was PACKED, and people were often turned away from parking areas due to lack of parking spaces. Plus, the roads are very narrow and windy. We didn't (but if I want back, I would) camp a few days on the east side of the park and a few days on the west side. It is quite a drive from one side of the park to the next.
Badlands-we stayed outside the park. Near Wall Drug. There were two or three campgrounds in the area. None of them were great, but adequate for a night.
Mt. Rushmore-one time we stayed at a private campground, maybe a KOA, which was nice. The last time we were there, we stayed in Custer State Park. We had electric hook-ups. We were close to a dump and water station. there were clean showers here, too. The park was great, and the campground was clean. In addition to Mt. Rushmore, I would suggest you visit Crazy Horse monument. We liked this better than Mt. Rushmore.