All I can offer is that I live in unit 31 and I never put in for unit 31. The folks I know that have good success out here generally have private land hook ups, or sort of just "road hunt" and drive the backroads. The exception to this is a couple of guys I work with hunt early in archery or muzzle.
What I've learned in my limited experience in hunting deer on the high plains desert is it's very easy for them to get spooked early in the season and run off to the safety of private land and is quite disappointing when you find lots of sheds, but only the infrequent doe during your hunt.
That being said, I'd suggest not bothering with the area north of Milnsand and concentrate south of there, the folks without private land access that I know who have success, seem to have it there. Check to see if there are any ranchers participating in the NM "open gates" program, and Good Luck out there!