With the retirement of the Space Shuttle program, National Geographic has put an article on their website they originally published in 1981. It features a photo of the Space Shuttle Columbia being towed through Lancaster, California (my home town) to Edwards Air Force Base.
The Shuttles were all built in neighboring Palmdale at the Rockwell Corporation facility at Plant 42. Each one was towed to Edwards where it was put on the back of a specially configured 747 and then flown to Kennedy Space Center in Florida.
Never to Texas.