I've hunted north of the areas you are considering for the last two years. Quite a few folks camp on public land without issue, but I wouldn't leave a gold brick (or your guns) out in the open. You never know. The Rio Grande corridor is notorious for illegals and drugs, but generally they don't get very far away from the main travel routes. I suggest you make your life a whole bunch easier and go the NMGF department's website and download their "CarryMap" application. It is updated annually showing landownership, roads/routes, fences and water for the entire state. It isn't fancy but I've found it very valuable and it is a download so you don't need cell service.
A recent observation, and this happened last year, though a bit later (early February). The Gambel's disperse away from their wintering areas. They move out of the draws and go somewhere. I have no idea where, but right now, quail hunting in this area is tough. I walked 4 days in areas that had enough sign to look like a quail farm a month ago, without seeing a track (sandy bottoms of draws/arroyos. Even the covey that hangs around the RV park we stay at is gone. They'll slowly start drifting back, but I haven't figured out where they go (yet). There are a few scaled quail in this area, but not enough to determine if they are doing the same thing.
Good luck!