The mother has led me to believe I'm the father, but is not allowing any shared custody or visitation.
Family Law Attorney
You need to file a lawsuit to determine parentage. The court will order DNA testing that you will have to pay for. Or you can go to the attorney general's office in Houston (there are several locations). Tell them you want visitation and will pay child support but you want DNA testing first. They will cover the costs. However, because they have so many cases this may take several months.
I recommend that you consult with an attorney regarding this matter.