For raw land, I like Landwatch.com. Be advised, some land is only good for rattlesnakes and rodents. A forested plot can supply all the lumber needed for a cabin.
To avoid pissing off your neighbors, I’d suggest a minimum 20 acres for shooting, especially if your cabin will be unoccupied for weeks at a time. If you are trying to stay small, look at the adjoining properties to see if your cabin will still be far enough away from existing homes, and that there is a home on each property, which will lessen the chances of someone building close to your cabin. If you get just one acre, your shooting will be very limited. When I last researched the state regs, it only prohibited shooting closer than 300ft to an inhabited dwelling, or at dwellings/vehicles/people etc.
I believe the ordinances for utilities are county specific, with rural areas being less restrictive. The poorer counties often don’t have the resources to enforce them, and people with land do as they damn well please, building codes be damned. Most everyone is armed, so most govt employees, including police, are reluctant to stick their nose in other people’s business.
Lots of counties will stick you with fees for trash pick up and other services even if you never use them, and if you don’t pay they slap a lien on your property. Usually, as soon as you get any sort of utility hooked up, that company notifies the county, which starts billing you for trash and other services. If you stay off the grid, you might escape their fees.
You can get beautiful weather in the central part of NM, The east mountains enjoy a southern latitude with high altitude, which creates a fairly mild climate, without the extremes of nearby Albuquerque.