You are not going to like my answer. So take a deep breath & start reading.
I've answered your question many times over the years on this website & on my blog - shown below.
Your husband is going to pay child support. It does not matter is he gets what is commonly known as "joint custody" in the State of Texas. What most people call JC does not mean that the man does not pay child support -- many people think it means that you don't pay child support - but it does not mean that at all.
I recommend that he talk to an experienced family law attorney immediately and find out about Texas law.
In Harris County, I recommend attorney Patricia Bushman at 713-807-9405. Call her office and make an appointment. I think her first appointment is free. She will accept TX A G cases. Many attorneys won't accept these cases because they are a real pain & the TX A G office is so backed up that they can take an attorney out of the office for 1/2 a day. Pat will still accept them. I rent space in her office & I think she is a really tough & smart attorney. You can tell her that I sent you to her.
If DNA determines that he is the bio. dad then he gets the rights of the father & the obligations of a father. And one of the rights & duties of the father is to support this child.
Ok - here is the part that you are not going to want to hear. But you need to think about this -- As a step-parent, you have very little input into the parenting of this child.
These parents created this child together & they are going to get to learn to parent this child together for the rest of this child's life. You get to observe and watch them do it.
If you want to keep your marriage together then I encourage & advise you to support your husband to be a great dad. I also encourage you to keep your mouth shut & to a step-parent. This is not your child. You were not there when the child was conceived and this is not your bio. child. If you are a good step-parent, this child will grow to love you. If her mom is not a good mom, then eventually this child could come to live with you. However, you need to take things slowly. You need to not try to be her mom. I have seen many step-moms eventually turn into wonderful "moms" but don't try to do it overnight. Remember, your place -- you are not her bio. mom -- you are her step-mom -- don't try to be her mother -- take things slow -- this is going to be very difficult for this child & she does not need any drama of two women fighting right now! She needs calm & people who love her in her life right now.
If he refuses to take the DNA test, then the court can just "find" him to be the bio. dad and order him to pay child support.
I would talk to an attorney immediately and learn the harsh reality about having one-night stands and unprotected sex. It does not matter if she trapped him or lied to him or whatever. There is no an innocent child born of this affair -- and that is all that really matters at this point. How this child came into existence really does not matter at this point -- all that really matters NOW is that there is a little child here now.
The bottom line line is that the taxpayers of Texas do not want this kid on welfare. The Legislature does not want any more kids on food stamps or on welfare and the taxpayers want bio. dads pay pay child support. From your perspective, this kid comes before you, any kids you have and any expenses you incur. So before you get pregnant or spend any money, recognize that this kid comes before any future kids you have or any future expenses you incur. It's not fair -- but it's the way it is.
If he does not pay his child support his credit report will be ruined, his income tax refunds will be grabbed and his bank accounts will be seized. Your car insurance will go up and he could go to jail for non payment of child support. The list of bad things that could happen can go on and on.
You can read my blog for the list at www.taxfamilylaw4u.blogspot.com -- it's just not pretty.
