I just returned from a quick trip to Phoenix. Stayed at the Oak Flat CG just off US 60 East of the Phoenix area, there are some backcountry areas but we just stayed in the CG, no fee, nothing fancy but nice sunsets
Further East of there US60 turns to US70 heading East, stay away from I10, trust me this is a better route
There is camping near Globe but we opted to head towards the Black Hills Backcountry Byway
http://www.blm.gov/az/st/en/prog/recreation/autotour/black_hills.html
We stayed at the campground, one other person there at the other end. The South end of the Byway is around 8 miles North of US70 off 191. If you go in there there is a good spot for a camp about 3 miles in on the right by an interpretive site, just a nice big spot 100' off the road with nice views. We didn't know about this and it was getting late so I opted to head in from the North end knowing that there is a CG about 5 miles in. Decent sites, I don't think any water but a covered picnic table and a primitive toilet. The CG is right above the Gila River and in the Spring the Cottonwoods should be out and very pretty.
Here is the scene at the CG
Pretty good sunsets here as well
The road is near 21 miles and shouldn't require any FWD unless there is a lot of rain, great views, nice picnic spots, some places would be tight for passing someone coming the other way.
This is a very nice and easy tour of the desert transitioning from Sonoran to Chihuahuan.
Are you familiar with some of the TX options? I can point out some areas here in S. NM and on the way to Fredricksburg. Balmorhea, Fort Davis, Amistad.
We think Big Bend might be nice but, otherwise, we are not familiar with TX options and would love to hear about your suggestions in S. NM. Thanks so much for your help and interest.