Secluded is a subjective term. I would not consider any public or private campground secluded. Those that accept reservations generally do so because they are popular and the management wants to give opportunity to all. New Mexico State Park campgrounds are generally not parking lots, but I would not consider them secluded. As mentioned, some NM State Parks like Elephant Butte and Caballo have primitive camping areas where you may be able to get away from other campers. Some New Mexico Wildlife Management Areas allow camping. You can find
National Forest Campgrounds in New Mexico on the website in the link.
The term dispersed camping is often used by RVers to describe what agencies often consider car/vehicle camping which can only be done in areas open to vehicle traffic while dispersed camping/back packing can be done in areas closed to vehicle traffic. On National Forests, areas open to car/vehicle camping are shown on the Motor Vehicle Use Maps (MVUM) or Travel Maps. These areas can change as often as every year and are not marked on the ground, so you need the maps. Maps can be obtained from the Forest Service Ranger Stations and many are available on line. Search for MVUM or the individual Forest's websites.
BLM lands have been mentioned. Be aware that some BLM areas like the Rio Grande Del Norte National Monument along the Rio Grande River restrict car/RV camping to developed campgrounds only.
Safe travels!
Trailertraveler