There are plenty of
Boondocking places in New Mexico, which can be found on Ioverlander. If you are looking for campgrounds, RV parks, etc. use google maps. There are numerous national forests with campgrounds also. We stayed at Coronado campground, a municipal campground, just north of Albuquerque, which was reasonably priced.
In Albuquerque make sure to take the drive, without the trailer, up to Sandia Crest. Do this on a clear day. You won't be disappointed. Bring a jacket.
if you are interested in birds, go south of Albuquerque on I25 and stop at Bosque Del Apache national wildlife refuge just south of San Antonio, NM It has a nice drive and visitor center. The quail at the visitor center are fun to watch.
If you take US 380 from San Antonio, NM east to Roswell on your way to Carlsbad Cavern, stop at valley of fire (GPS I33.685, -105.919), which is an interesting volcanic area and campground.
Also in this area is the VLA, which is a radio astronomy site on US60 west of San Antonio, NM. I have not yet had the opportunity to visit it, but it looks interesting.