Legal Question in Family Law in Texas

Yes, she certainly can--you are now their father, for all intents and purposes, including taking on the responsibility to support them. Whatever the biological father's paying now, I would assume is probably making up unpaid back child support, not paying for them currently. If you weren't prepared to take on this responsibility, there was a time and a place to say that. It's too late now, because you are now the only father they have to rely on for all of their current and future needs.