We have stayed at Cave Country RV campground in Cave City KY. Its about 10 minutes or so from Mammoth Cave. We would go back for sure. The down side is that there is very little on the campground to do. No playground, pool, little shade, etc. On the plus side all the sites are pretty level, gravel, and pull through. And there is a new public park/playground within walking distance. The crowd was a little older with some kids around, so it was pretty quiet. The hook ups were strong, and the workers were extremely polite and helpful. We stayed on the opposite end, but FYI there are train tracks that run right next one end.
We drove through the Jelly Stone park and there was plenty to do for sure but man was it crowded and loud. I have heard you can get a day pass to play even if you don't stay there so that be worth looking into.