First a note about Rushmore. We stayed at a national forest campground at Sheridan Lake and it was really nice. Water and dumps are available and you can run generators at certain times. It is close to a lot of the tourist activities and yet fairly off the beaten path. We much prefer national park, national forest, COE, and state/county parks to private parks because they seem much more natural to us.
We were at Yellowstone this past late summer and stayed at Indian Creek campground. Nice relatively primitive campground with no water or dump and I don't think they allow generators so have water tanks full and black/gray empty and batteries charged. We moved up to Grant Village after two or three nights at Indian Creek. Grant Village is huge with water and dumps and you can run your generator there.
We went to Tetons after Yellowstone. We actually saw a lot more wildlife at the Tetons than at Yellowstone. We stayed at Gros Ventre campground. Had moose with babies there in the campground a couple of mornings. Gros Ventre is reasonably close to Jackson yet easy access to GTNP as well. It was pretty full even in fall when we were there.
I hate making reservations because it ties you down. I prefer to drift at my whims. Usually you can get a site if you show up at these park campgrounds right around 10:00 in the morning when others are vacating. Sometimes this means you have to get up early and move from a previous site but it usually works for us.
Have a fun trip.