Help in Harris County!!!

I need advice on the possibility of being granted a step parent adoption. My situation is as follows: I had 2 children with my ex-husband in 2008 and 2011. He moved back to Ohio in September of 2011 when we separated. My youngest was 2 months old, and my oldest was 2 almost 3. He has not seen them since. He was ordered to have supervised visitation with the SAFE program because of domestic abuse and child abuse. I never called the police on him because of fear at the time, but I have witnesses to the abuse and his own confession in messages to me. When he left he wanted nothing to do with the children. He did not start paying child support (automatically deducted from his paychecks) until 2013. He has fallen behind several times, has filed a motion to have child support dismissed (which was denied), and the CS office recently began a case against him for failure to pay. He started paying again (once they discovered he had employment) but is still $2,000 behind. I have since had 2 more children with my husband, who has been in their lives for the past 4 years. My 2 oldest children only know him as daddy. My oldest does not remember his biological father anymore, and because of emotional issues he had following the abuse I have not reminded him. My husband has loved and cared for them as his own. He wants to officially adopt them and move on as a family. My ex-husband has made no effort to see them and has continuously tried to get out of paying child support, but I am fairly confident that he would contest the adoption out of spite. He is not mentally stable, had been on medications previously, and has threatened to attempt suicide previously. My question is this: do we have grounds for abandonment even though he has paid child support periodically? Also, just opinion of course, is there grounds for termination and granting of a step parent adoption?