It may seem that in today's world there are too many men trying to avoid fatherhood. But this may just be that those who do so are getting all the press. Good men who want to meet their obligations and share in the joys of parenting will want to establish their legal rights over their children.
If a man thinks himself the biological father of a child, he has the right to file a lawsuit to prove that. The case is called a "Suit Affecting the Parent-Child Relationship" (SAPCR). Even if the mother has a husband (making him the "presumed father") a man can challenge the paternity in court.
Texas has set up a Paternity Registry, the purpose of which is protect the parental rights of fathers who want to assume responsibility for children they have fathered. The registry is run by the Texas Bureau of Vital Statistics.
So let's suppose a man thinks he may have fathered a child. He is a stand up guy and wants to meet his responsibilities, or perhaps would welcome the joys of fatherhood. He can file a registration before the birth or within 31 days after the date of birth of the child. If he fails to do this, should the mother decide to give the child up for adoption, he would not receive any notice of the proceedings and could have his paternal rights terminated without his knowledge. If he registers however, he must be given notice of any lawsuit to terminate his parental rights and have the child put up for adoption. Once he knows this is going on, he can join the suit and file his own claim of paternity so a judge can official rule that he is the father and has the rights of a parent. If he does this, and the judge rules he is the biological father, the mother cannot have his rights terminated on her own, even if she wants to give up the child for adoption or someone else wishes to adopt the child. The father would have to either voluntarily give up his rights or be found to be an unfit parent to have his rights involuntarily removed.