Our favorite state parks anywhere are in New Mexico. Just this year, we stayed at Sugarite Canyon, Heron Lake, and Santa Rosa Lake. In past years, we have also been to Elephant Butte, Caballo Lake, Percha Dam, Clayton Lake, Storrie Lake, and Navajo Lake. Most are on Corps of Engineers or Bureau of Reclamation lakes.
You may not be able to stay a month (most have a 14-day limit), but it's hard to beat $14/night for W/E sites with a dump, flush toilets, and showers. In SE NM, there are Brantley Lake north of Carlsbad, Bottomless Lake at Roswell, and Oliver Lee at Alamogordo. You will likely need reservations at Oliver Lee, as it has only 18 electric sites and is the closest SP to Alamogordo, Holloman AFB, and White Sands Missile Range (ie., lots of local RVers!).
"Retirement is the best job I ever had!"
2016 Rockwood Signature Ultra Lite 8244WS 5th Wheel; 2015 Ram 2500 4x4 CC-SB 6.7L CTD
Sold: 2012 Rockwood Mini Lite 2109S