We track live availability at 7 campgrounds with cabins in New Mexico, across 6 towns. A cabin is the hardest thing to get in most park systems: there are only ever a handful per campground, they book the day the window opens, and they stay booked through the whole season. When one comes back it is usually a cancellation, and it is usually gone within the hour.
Booking goes through Recreation.gov. Searching is free.