Almost everyone I know from Texas has an intense twangy drawl, but then again, almost everyone I know from Texas is from the Southeast. A friend of mine from Arlington has no real accent, and a friend from El Paso has a slight California-type intonation, which I call Californian but is really a slight bit of Spanglish. I think North and West Texas don't really talk with the drawl like the South and the East.
Once we were talking to a shopkeeper who was uncanny with accents. He had my grandmother nailed within the first few seconds she spoke ("Green Bay Wisconsin, give me a break," he said) and even my Dad, who I thought has lost his Texas accent, was easily put in the category of "What part of Southeast Texas are you from?"