Our favorite campground in NM is Valley of Fires, near Carrizozo. This well-maintained campground has paved roads and sites, covered ramadas, BBQ grills; and it was remodeled in recent years. 14 of 19 sites have water and electric hookups. Clean, modern restrooms have flush toilets and great showers. They also have a dump station. It's a NM State Park on BLM land, which honors the National Park Senior Pass discount of half-off camping fees.
GPS/Google Earth coordinates: 33° 41.089'N 105° 55.197'W
We also enjoy Bluewater State park, just south of I-40, near Prewitt exit. Gate is open 6:00 AM to 9:00 PM; and it's OK to park overnight outside entrance, if gate is closed.
It has paved roads, and some sites have water and electric. Dispersed camping on shoreline is $8, regular sites = $10, electric = $14, water & electric = $18. Dump station available.
For more info, see:
EMNRD State Parks Division
Tiger Muskies at Bluewater Lake State Park - YouTube
GPS/Google Earth coordinates: 35° 18.333'N 108° 6.340'W
We also enjoyed the one trip we made to Elephant Butte Reservoir, but I don't have details on that location.