White Sands dunes are not as big height wise as Great Sand Dunes but cover a larger area and you can drive into the dunes. People die hiking the trails in White Sands. If you are going to hike follow the advice to take 1 gallon of water per/person.
Oliver Lee state park is just south of Alamagordo and is reputed to be a good campground, although I have never stayed there. I did stop in Petroglyphs state Park 15 years ago but cannot say how it is today. Petroglyphs is north of Tularosa off of 54 and is a bit of a drive if you were planning to make that a HQ for your visit. Nice place to see though if you have time.
White Sands Missile Range HQ has a missile and atomic bomb museum that some find interesting. I would stop as you are driving between Deming and Alamagordo. It is about 5 minutes off of US70 to get to the main gate. The road is at the bottom of San Augustine pass and you can park outside the visitors center in the truck lot and walk (1/8 mile) to the museum, just inside the gate.