First of all I am not a tax person or lawyer and don't play one on tv so you shouid consult a professional in your areas.
I'm sure someone with more properties may be able to better answer your question but here is my experience:
I live in CA where we DO have state income tax and I have rentals in Ohio. I file my rentals on my federal & state returns but Ohio requires that you file an Ohio state return if you profit a certain amount (I want to say $3500/yr of the top of my head but don't quote me, I would need to go look to be sure). It's my understanding that here in CA I could offset on my CA return anything that I pay to Ohio but I haven't had to do that yet as my "on paper" profit has been below the minimum after write offs. I'm assuming (never assume, lol) that since you don't have state tax in TX you would just file an Ohio return if you make over the amount required to file.
Also, the state tax for Ohio would only apply to the income earned on the rental properties in Ohio, not any income you make in TX.
Again, no tax or legal advice given. Good luck and success on your real estate endeavors.