I highly recommend that you meet with an attorney in person & discuss your questions. Your husband is on the birth certificate as the father. Please talk to an attorney. Look on this website & find one that you like. This on-line site is not appropriate for addressing your concerns.
You should really speak to a licensed Texas attorney about this matter. They will need more information about what happened such as: Were you married at the time of the child's birth? If not, did your current husband sign a Acknowledgement of Paternity? I am assuming this is the case if he is on the birth certificate. How long have you been living together with the child? These questions matter because the answer will tell us if your husband is the presumed father.
If your husband is the presumed father, your ex will need to file suit to prove that he is the actual father. If he succeeds, he may then be responsible for child support regardless of whether he is successful in getting visitation. Being a parent comes with both responsibilities and rights.
This is not legal advice and is only intended to provide a minimum amount of information for the purpose of helping you determine whether you should obtain assistance from a lawyer. I can not offer legal advice without knowing the full facts of your case. If you would like to speak with me further, please give me a call.
If you were married to your husband when the child was born, then your husband is the presumed father regardless of whether he is on the birth certificate or not. Your ex has four years to petition the court to establish paternity however you can also request that no DNA test be performed and argue that it is not in the child's best interest. Please contact an attorney to discuss further.
This is general legal advice. An attorney-client relationship has not been formed. Please contact an attorney for specific advice on your case.