First, just to establish where we are: Malvaviscus arboreus var. drummondii (Turk's cap or turkscap) blooms both white and red, and is native both to North America and Texas and, according to this USDA Plant Profile Map, also grows natively in and around Nueces County. Malvaviscus arboreus var. drummondii 'Pam Puryear' (Pam Puryear Turk's Cap) is a hybrid between drummondii and Malvaviscus arboreus (Turkscap), both of which are native and the hybrid produces a pink flower, but will not appear in our Native Plant Database.
St. Augustine grass is native to Africa, but in view of the amount of shade you have, it may well be the only grass that will survive. Unfortunately, in these days of heat and water shortages, it is high maintenance and requires a lot of water. We are encouraging the use of native grasses although, as we said there isn't much to recommend for shade. Had you asked in advance, we might have recommended a good quality mulch over those tree roots instead of the grass; however, it's a little difficult to unplant. On the subject of the oak suckers you initially were trying to eliminate, please read this previous Mr. Smarty Plants question.
Now, let's go on to the part of your garden we consider most important: the oak tree. This is a very valuable part of your landscaping and, although you didn't tell us what kind of oak it is, many oaks are threatened by Oak Wilt Disease. This is often spread by work done around the oak trunk or cutting the roots, resulting in a wound.
So, recommendations for your project? Don't cut any more oak roots. Remember that oaks possess the trait called allelopathy, in which they emit substances to discourage competitive plants beneath them. You could do everything right and the plants below that oak will still die. We think you are watering too much. If the dirt in the planting holes was not amended for drainage, that extra water may be just standing on roots. You haven't done a single thing that we haven't done in the past. What you do is learn from the results. Your plants may all do beautifully, the oak sprouts may die away and the oak continue in good health for many more years. But if they don't, just figure out why and don't do it again.