Can't help you with the Texas part, but can help you with general lawn tips.
In the heat of the summer, let the grass grow longer than you normally would - longer blades protect the ground from as much drying out. You can cut your lawn shorter in spring and late fall, but not during the heat. When you mow, don't take off more than 1/3 of the blade of grass - too much moisture loss. If you don't have a mulching mower, consider one. The shopped up clippings protect the surface of the soil from moisture loss, and provide free fertilizer. Water less frequently but deeper! If the roots go deep because the soil is wet down deeper, they are stronger. If people just water the surface of the lawn, the roots stay up close to the surface and then are more subject to drought. So on the days you can water, water one section much more thoroughly - an inch a week is great. Put a rain gauge or a simple plastic bowl out to measure what you've put out there via the sprinkler (just check it before evaporation kicks in). Water early or late in the day to limit evaporation. If you can aerate the soil by renting an aerator, great - but be careful about doing it now while the grass is under stress.
Finally, consider some raised vegetable gardens if you are allowed - grass is a cool/rainy weather plant that worked great in England but which, unfortunately, we've tried to import to the US with limited success and tremendous cost to both pocketbook and water tables!