Three people and a dog living in an Explorer in Texas in summer...wow I can understand what you are dealing with! My Explorer's stifling enough with just me inside, camping in the heat and humidity.
Unfortunately for you, Texas has an notable paucity of public land. I'm very lucky to live and work in Big Bend National Park, where the temps are very mild at 5400 feet (Chisos Basin, highest populated place in Texas). Unfortunately for those of you who live elsewhere in the state, there are no cooler elevations to climb into. If you make it out here though, you can stay for up to 14 days at a primitive roadside campsite in the desert backcountry. One campsite is over 4000 feet high and close to a beautiful wooded canyon. Entrance fee to the park is $25 but I could get you the campsite for free.
When I travel through Texas, I have never had a problem sleeping at the picnic areas located all along our two lane highways, but I doubt law enforcement would tolerate long-term camping at these facilities. If you have the funds though, you could slowly meander out here to west Texas and then on to New Mexico, where public land is abundant.
