RE: Anyone from New Mexico?
Your perceptions are correct, even the life long residents have trouble understanding the rules here. I moved here about 8 years ago from TN myself and it took me about 2 years to figure enough out to be able to hunt, with the help of a resident hunter. Most of the big game hunts are by draw and there is no prefrence point system like some states have, basically luck of the draw. There are some turkey and bear OTC tags but not much else. The other big difference is you can only get one tagfor each species, i.e. 1 deer, 1 elk, 1 pronghorn, etc., if you get lucky enough to draw. There are a few exceptions such as turkey where you can get a second tag for some hunts.
As for public land, it seems like half the state is public. Just in the Gila area there is something like 2 million acres of public land. Not all game units have significant public land but most of them have some amount of public land.
If you haven't been in elk country during the bugeling season you are in for a treat, there is nothing like being in the woods within a few hundred yards of a big bull elk screaming at another big bull elk. If that don't get your blood pumping nothing will.
What will be bringing you out this way? To what area?