I lived in NC for 2 years (many years ago), and I lived in the Washington DC area for about 5 years. My (home) is Indiana, I traveled back and forth a lot in those days.
We've also traveled from Indiana to Wisconsin, to Minnesota, to Iowa several times as well.
I will say, we have never stayed at a "bad" state park anywhere between Virginia and Iowa or Minnesota. Traveling that basic line, between DC to Indiana, then upward to South Dakota, just pick any State Park along the journey and you won't have a bad choice at all.
FYI, we are going to Mount Rushmore in July for a 2 week trip. We're really excited about going too. We've never been there.