First of all, Mr. Smarty Plants has had lots of question about controlling oak suckers under trees. Please visit the answer to a previous question for a good discussion about the merits of the various methods to deal with oak suckers.
There may be a problem growing plants under your oak tree, not just because it will be shady but because the oak tree has an allelopathic effect on some other plants. Allelopathy is a situation where a plant releases a chemical that has an effect on another plant. The effect may be beneficial or harmful, but the ones we hear about the most are those that are harmful to another plant. One of best known of the allelopathic plants is the walnut tree (Juglans spp.). It has a very strong effect on most plants attempting to grow beneath it. Trees with a lesser allelopathic effect include oaks, hackberries and eucalyptus trees. Red oaks, which in your area is likelly to be either Quercus shumardii (Shumard oak) or Quercus falcata (Southern red oak), are mentioned more often than other oak trees. You can read more about allelopathy in this article, Allelopathy and Arboriculture: Theory to Application. That said, not all plants are susceptible to the oak's chemicals. Some plants can grow under oaks. I couldn't find a list of plants native to Texas that reportedly grow under oaks; but, by observation, I can tell you some plants that are native to both South Texas and Central Texas that will grow under oaks.
Malvaviscus arboreus var. drummondii (Turk's cap or turkscap) grows in part shade and shade.
Lantana urticoides (Texas lantana) grows best in sun.
Sabal minor (Dwarf palmetto) grows in sun, part shade and shade.
Lonicera sempervirens (Coral honeysuckle) will trail across the ground (as well as climb) in sun or part shade.
You can check our list Texas-South Recommended for other possibilities. The plants on this list are native to your area, commerically available and suitable for landscaping. You may have to experiment a bit to see if the plants you choose will thrive under your oak.