Need Advise! Step Parent Adoption in Texas

Hello! I am new to this forum and I am just beginning the process of looking into Step Parent adoption in Texas. A little background. I have 2 children from a previous marriage. My son is 11 and my daughter is 9. My ex h and I were divorced in 2008. My DH and I have been together since 2009 when my son was 5 and my daughter was 3. I have sole custody of my children. My ex h does not have visitation rights. He was abusive during our marriage and went to prison for 4 years for assaulting me right before our divorce was finalized. He has not seen my children in 6 years and has only spoken to them on the phone once about 6 months ago. He has never paid child support and is about $30,000 in arrears. He has no interest in being apart of their lives, yet he refuses to sign over his paternal rights because he wants my children to have his last name. He is still currently a heavy Meth user and has been in and out of jail and mental institutions for Bipolar and Schizophrenia. Ok... now, my current Husband. We have been together for almost 6 years. He is an amazing man and adores my children. He is their Daddy. He is the only father that they have ever known. My daughter doesn't remember her biodad at all and my son wishes that he didn't remember. The only real issue that we have would be my DH's criminal background. He was a dumb and misguided kid who made a ton of mistakes and spent 7 years in prison for them. Since he has been out (especially since we met and got together) he has turned his life completely around. He is a wonderful man and a great provider. He is in the oilfields and allows me to stay home to raise the kids. Does anyone know if my DH's criminal background will preclude him from being able to adopt my children?