You can only stay on BLM land for fourteen days at a time, unless you pay for the Long Term Visitor Areas in SoCal and AZ.
My experience has been that I'm usually about ready for a change by then anyway, so I'm not trying to be a Debbie Downer.
You may only have to move 20 or so miles away, so do your research on that. I ran out of food and money in Quartzsite and was able to find enough wild mustards and other greens to survive.
The food bank just had stale candy bars and presweetened children's breakfast cereals.
I busk, so this was not my only source of food. Please stay hydrated if you are going to be near the Rio Grande and you might need electrolyte supplements to keep your brain from going wonky.
New Mexico State Parks Pass is <$300 and something you also might want to consider. A lot of my older and disabled rubber tramp friends spend the summers in NM State Parks and the winters at the LTVA.
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I am planning on camping long term somewhere along the Rio Grande in south New Mexico, and ideally to sustain myself mostly from the land. Does anyone have experience with this or similar?