Without oil, and run by their typical rich redneck pols, it would die down into the trailer park scenario. The presence of big oil money hides their governance flaws very well.
With competent people in place, their (things I didn't mention prior) massive aerospace/defense industry, substantial auto manufacturing plants, major seaports & general rail and truck shipping hubs, and impressive tech & communications industries could be beefed up even more to compensate for that missing oil revenue stream pretty well. It's all a pipe dream, though. I can't imagine Texas being governed competently, efficiently, and fairly enough to thrive up to it's true potential any time in the next 75 years, whether it remained a state in the union or not.