This app was mentioned in 5 comments, with an average of 3.60 upvotes
Some bus dwellers made an app called US Public Lands the works quite well. It's a few bucks I think - well worth it to save you the hassle of finding places on various sites.
Here's the android link, I'm sure there's an iphone version too. https://play.google.com/store/apps/details?id=com.twostepsbeyond.uspubliclands
BLM or forest service 99% of the time...but yes.
also download an app to see what the land is. Something like this
https://play.google.com/store/apps/details?id=com.twostepsbeyond.uspubliclands
Might want to load the BLM Rec Map on wifi; it's soaking my slow data plan.
I also use the US Public Lands app. By nomads for nomads! Works offline after initial installation. You can have it one while driving and it will show you when you are near public lands.
Look into this
https://play.google.com/store/apps/details?id=com.twostepsbeyond.uspubliclands
An app listing US public lands, should be OK to camp.
NM averages over 4k' of elevation and is rather cold in the winter. It's popular with the ski folk. Very popular for summer.
TX doesn't have a lot of free public land, although I have stealth camped in many TX cities. There is some free public land down on the coast I have not explored and probably won't since it's so far from other free options.
AZ is a common wintering spot for snowbirds. Most van folks boondock on federal land as /u/lennyflank says, and RV folk either boondock or rent an RV campsite. There are state lands, but the camping allowance is very limited, non-free, and carries a $1000 fine for the unpermitted. I use the US Public Lands app to make sure I am on federal public land.
There are LTVA in AZ/CA where you can pay $180 to camp for a long time in one spot. I've never done it but I might try it this year.