He is correct. The court has jurisdiction over the child but not your ex. The court can decide who and where to raise the child but the court cannot force the guy to pay for child support, unless the guy has some kind of minimum contact with TX that the allows the court to find jurisdiction. I am not familiar with TX law but this analysis is based on constitutional analysis. I doubt the TX court will assert jurisdiction over your ex only because he wrote a letter to the court. Unless he unequivocally consents to the TX jurisdiction, minimum usually means much more than that.
What your attorney should do is to first, establish paternity of the father; second, use child custody as bargain chip to get child support from him, understanding you may never get 100% custody and may never get this guy out of your life until the child reaches majority. Same thing goes for the uninsured medical expenses.