On your way to Arizona on I-10, you might consider a detour to visit Carlsbad Caverns and Guadalupe Mountains national parks. If you have a Federal Senior Pass, you can have inexpensive dry camping in lots of BLM and Forest Service campgrounds in southern New Mexico, like those in Lincoln National Forest. You can also dry camp in "dispersed camping" areas. Here is what the Lincoln NF website says:
"Dispersed or "road-side" camping is allowed throughout the Lincoln National Forest. You may drive to park your RV no more than 300' from an open road designated for such use. Please check with one of our offices to obtain a free Motor Vehicle Use Map which details exactly what roads you are allowed to disperse camp along. Most roads, outside of US and State Highways are available for this use."
One of our favorite area for dry camping ("boondocking") in that region is the Sunspot area just south of Cloudcroft (west of Carlsbad on US Highway 82). The views of the Tularosa Basin, Organ Mountains, and White Sands are incredible from Sunspot. I would recommend this area for November or April, as they do have skiing in the winter in the Cloudcroft and Ruidoso areas.
You can also dry camp in dispersed camping areas in the brand-new (May 2014) Organ Mountains Desert Peaks National Monument, located just east of Las Cruces. However, the only developed campground in the monument will not handle motorhomes or trailers over 23 feet long. Also nearby is White Sands NM, which you shouldn't miss if you are in the Las Cruces area.
Finally, if you get tired of dry camping and want hookups for a night or two, the best deal in southern NM is the BLM's Three Rivers Petroglyph Site north of Alamogordo. If the water has not been shut off for the winter, your Senior Pass could get you one of only two electric and water hookup sites for $9/night. And the fantastic petroglyph trail is a bonus!
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2015 RAM 2500 4x4 crewcab 6.7L CTD; 2016 Rockwood Signature UltraLite 5th wheel