There are NFS campgrounds in the Cibola National Forest not far from Pie Town. If you stay in Grants and Gallup, get full hook ups. It's hot. But there are some amazing things to see in the area, especially around Grants with the old volcano, the ice cave and all the sand stone bluffs. You could consider staying around Chama or the Red River area. We will be camping just east and north of Chama in a few weeks at a NFS campground we really like.
I grew up in New Mexico and absolutely love the state. Their green chili cheeseburgers are a staple for us.