You have the option of going Brownfield, Artesia, Alamogordo. That takes you over a really nice route over the Sangre de Christo Mountains through the 9,000 foot high Village of Cloudcroft and the nice run down the west side to Alamogordo and White Sands. If you don't regularly get to see some heavily forested mountain scenery, it's a nice trip. A bit of a longer trip than the Carlsbad route, but much more scenic.
White Sands National Monument is there, of course, There is a picture postcard old timber curved railroad trestle just west of Alamogordo with a pull-off overlook. If you're really taking your time, the National Solar Observatory is at Sunspot, south of Cloudcroft.
The drive is about 6 hours of you go on 62 and 180 through Hobbs New Mexico that route takes you though Carlsbad and it only a slight detour to the famous Carlsbad Caverns. It would take the elevator down it will still take at least four hours to take a tour of the cavern, but you could do that though it would be a long day when you add time for gas and food stops