If you want to go south: Here are some neat places in NM and AZ if you have not noticed them on the forum yet.
Texas State Parks are really nice: http://www.tpwd.state.tx.us/spdest/findadest/
South Llano River State Park adjoins Walter Buck Wildlife Management Area, south of Junction is great as is Pedernales Falls State Park in Blanco County east of Johnson City. We have been to both. http://www.tpwd.state.tx.us/spdest/f...h_llano_river/
Balmorhea State Park is located in the foothills of the Davis Mountains southwest of Balmorhea and nearby Davis Mountains State Park (along with the famous Big Bend Nat'l Park) are very nice places and one could spend a long time at each. Been to all except Davis, but it is nice areas, be sure to visit the old Ft. Davis a nice place to see.
Goose Island State Park, surrounded by the St. Charles and Aransas Bays, is located north of Rockport & near Freeport in Aransas County. Nice area by the Gulf of Mexico. I want to do this one someday.
Mustang Island State Park with about 5 miles of beach on the Gulf of Mexico in Nueces County, south of Port Aransas. We liked this one for the beach.
Lake Corpus Christi State Park southwest of Mathis was a nice one. But not very close to Corpus. http://www.tpwd.state.tx.us/spdest/f...orpus_christi/
Goliad State Park is located by Goliad with a nice campground on the San Antonio River. (a lot of Texas History in this area, some from same time as the famous Alamo fight: http://www.tpwd.state.tx.us/spdest/f...spiritu_santo/
) Been to it and would go back.
Falcon State Park is located north of Roma at the southern end of the International Falcon Reservoir in Starr and Zapata Counties. A lot of snow birds winter in this area.
Lake Mineral Wells State Park is located east of Mineral Wells in Parker County. I would like to visit it someday. http://www.tpwd.state.tx.us/spdest/f...mineral_wells/
Seminole Canyon State Park and Historic Site is located west of Comstock near Langtry in Val Verde County. I have this one on my to see list someday. We have stopped at the Langtry Visitor Center. I love the history of the area.
Monahans Sandhills State Park is about a half-hour's drive west of Odessa. A neat place.
Big Bend National Park is also a choice. http://www.nps.gov/bibe/
Guadalupe Mountains National Park is a possiblity as well. http://www.nps.gov/gumo/
Padre Island Nat'l Park could be also. http://www.nps.gov/pais/
I realize these would have to have a lot of driving time invested to get to them and return. Maybe they would be better once you have a month or more to invest into a trip. Good luck in finding a somewhat dry place, however a lot of the places a little south of you may be colder than were you are. At least as I understand the weather up in your part of the world.
Good luck! We tend to be high and dry here most of the time. We were lucky each time we visted your neck of the woods to be in a dry period.