I don't think there's anyplace that's unsafe. Just drive until you're tired, pick a place and spend the night. There aren't a lot of choice on I-10 in west Texas or New Mexico. Just aim for a town and they're spaced far apart.
For New Mexico, Las Cruces or Deming are your best bets. Arizona: Benson - Cochise Terrace is good or Kartchner State Park and tour the awesome caverns.
Even if you're not a member, a great place for a week is to use the special $50 offer for a full hookup site in a beautiful, fun park in Benson called Escapees Saguaro Park. Huge sites and friendly people. Lots to do in the area - Tombstone/Bisbee/Kartchner Caverns.
http://skpsaguaro.org/special-offers.htmlTexas: Fort Stockton and Van Horn have parks.
