Your lawn sounds like mine.
I removed a large Catalpa (Indian Bean) tree about 3 years ago. When the lawn was shaded, it was St. Augustine, and it was nice. Sometime before I bought my house someone must have seeded Bermuda, because it is taking hold now. With full sun and the drought we had last year, it looked scorched.
I aerated it, because I had the old dead St. Augustine thatch everywhere and then I scattered red sand and green sand on top. North Texas Soil isn't really soil, but clay, so the sand breaks it up. Without any watering from me, my lawn is still green. We are still under drought restrictions, and I use the alloted water time to water my vegetable garden and soak the foundation.
I would suggest doing a soil sample before you do anything else, because if you spend money and time trying to do something to crappy soil, you have just lost your money and time.
I get a lot of my info from Texas A&M's ag department. You might be able to get info from the University up there. My cousin is a prof there, I can find out if their ag departments has anything published.