I live in SW New Mexico. Big difference between North and South in the spring. Surprise snow more likely in the northern part of NM. Yet, we have had snowflakes as late as May 4th. It's not every year, but it can happen, especially with climate change. (wink, wink) Weather is the main worry in our spring.
State Parks are generally uncrowded until school lets out in last week of May. We have stayed in many across the south in the spring. Our worst experience was at Brantley Lake with afternoon winds over 50 mph. Luckily they calmed down after sunset. In fact, winds are most fierce in the spring all over New Mexico. Many warnings for high profile vehicles.
Our state parks are usually at some interesting place. Nearby we have the "City of Rocks" and "Rockhound" parks. You have probably investigated the state park website, but if not, here is a link to the map of the parks:
New Mexico state parks mapCome and enjoy our state.
Rick