You should not necessarily beat yourself up over letting the work proceed without getting everything signed up in advance. In the real world, that happens all the time. It doesn't mean that you are without legal rights.
Even though you haven't signed the contractor's proposed contract and he didn't sign your proposed addendum, you still may have a valid and binding "oral contract." It depends on whether there was a meeting of the mind on all of the essential elements of the deal, like what work was to be done, how it was to be done, by when, and for what price. My guess is that you probably DID have a valid, binding contract, although what you understood the terms to be and what the contractor understood the terms to be may be in dispute.
Keep your receipts. The fact that you don't (yet) have any acknowledgment from the contractor that you've paid in full (between what you've paid him and the subcontractor) isn't crucial -- not yet, anyway. You didn't mention any attempt by the contractor to try to collect more from you, nor any liens that he might have placed on your house in an attempt to force you to pay. If he knows that there were problems with the work, then that may explain why he's not trying to collect anything else; or perhaps he actually has been paid by his subcontractor under whatever deal they had with each other.
If indeed the contractor gave you instructions to pay the subcontractor directly, then yes, you ought to be entitled to credit those payments just as if you had paid the contractor.
In general, the contractor is responsible for unsatisfactory work done by his subcontractor, just as if he had done that bad work himself. He had the duty to direct, supervise, and inspect. It sounds like he didn't. You should be able to assert your claim against him, without having to try to chase down the out-of-state contractor.
By "assert your claim," I mean, first, presenting it to him in written form. I'd suggest that you write down exactly what happened, what you are dissatisfied with, why you're dissatisfied, and what you want to see done to fix it -- either in terms of him making repairs himself or, if you no longer trust him, what you believe he needs to pay you to have someone else do it competently. Send him that by certified mail, return receipt requested, with a duplicate copy by regular mail (because sometimes people don't pick up their CM/RRR, but there's still a presumption he got it if you sent it regular mail and it didn't come back). That might well be enough to get him to satisfy you, but if it isn't, you'll need to decide what to do next.
You might be able to get volunteer legal help from one of the clinics at one of the local law schools (University of Houston, South Texas College of Law, or Texas Southern Law School), or through the Houston bar. Depending on how much money is involved and how solvent the contractor is, you may also be able to find a lawyer (probably a young hungry one) who'd take your case on a contingent fee.
But you may be able to represent yourself in small claims court, especially if it's less than $10,000, without a lawyer. From the way you drafted your questions, I'm guessing you'd do a fairly good job in presenting your position to one of the Justices of the Peace who preside over small claims courts, and they're pretty savvy. I'll link below a pretty good article about small claims court proceedings that might help you.
Whether you have a lawyer or not, you also may want to consider bringing your suit not just for breach of contract, but for violation of the Texas Deceptive Trade Practices-Consumer Protection Act. You may be able to recover additional types of damages for that (for example, mental anguish damages if the contractor's breach of his implied warranty of good and workmanlike performance was intentional).