Like eubank said, some forest campgrounds may still be closed at that time of year. The national forest websites post opening dates. If you get time check out Capulin Volcano Nat Monument, and Fort Union, right off of I25.
North of Las Vegas NM there is Storrie Lake State Park.
Bandelier National Monument, has a nice campground. From there make a day trip to Los Alamos and visit the Bradbury Science Museum, https://www.lanl.gov/museum/ free admission!
Check out the Sacramento Mountains in central NM. Ruiodoso, Cloudcroft, WhiteSands Nat Monument. There are lots of trees in the central mountains. For fun go to the UFO museum in Roswell, and don't forget Carlsbad Caverns. West of Carrizozo is Valley of Fires Recreation Area. All these can be found on the web.
Full hookup NM state parks are $14.
April can be very windy in New Mexico.