Check out the tunnels along highway 16, they line up with the monument. Stop at the second or third one, because the first one (from each direction) is always very crowded. You can take a picture of the monument and the tunnel all around the frame of the picture.
Also when leaving the monument, there is a tiny parking lot to the right, it gives a side shot of Washington's head.
I love dry camping, and stayed at Custer State Park. No hookups though. It was near a self guided tour, you can take in a car, to see wildlife.
Fred.