Alamogordo has a very nice space museum that is worth the visit. Also the US Army base at the west end of White Sands on US-71 has a fantastic display of missles and such tested at the range.
We've overnighted in Tatum. Not a four star park, but heck, it's free and comes with 50 amp full pickups in a safe area. Nothing fancy, but clean and people were nice when we walked into the corner café for breakfast in the morning.
Amazing amount of metal art signs. Little has to win the award for best street signs in all of the United States.