While I love the Badlands, and make it a point to overnight there on my travels west, it is pretty much a drive-through and stop at scenic pull out type of place. Of course there are trails and hikes and interpretive displays - but you can do all the stops in less than a day. Roads are great, but you will find limitations on which pull outs to stop at due to the size of your rig. The park campground is pretty neat, with great ranger led evening programs, but no full hookups. But check site sizes and reserve ahead. It is desert like, so don't plan on any shade.
Just outside the south park entrance is a KOA, near the town of Interior. Well treed on the banks of a river. Very, very nice - and they would have no issue accommodating your rig and providing full hook-ups.
On our last trip through, we pulled into the KOA in the late afternoon, then took the drive through the Badlands the next morning pulling our 30' fifth, stopping in Wall for lunch. Final destination was Devils Tower for the night. But having been through the Badlands a lot, we didn't stop at many of the pull outs.