Child parentage issues

My ex boyfriend and daughters bio dad is waiting until last minute to file. He is literally waiting til the day before the filing date which is February 12. I left my daughter's bio dad at 5 months pregnant. I found out I was pregnant in November then I found out he was a sex offender who is registered on Megan's Law. He raped his own sister I don't want him around my daughter. The minute I found out I started looking for shelters and I found one. I was going to give up my daughter for adoption but chose not to her bio dad was not happy that I chose to keep her. He is not on the birth certificate. From the beginning he wanted nothing to with her then last year he wants to be involved all of sudden now that I'm married. Well I met my now husband in July he has been nothing but good to me he took in my daughter as his own. My daughter and husband have a great relationship. My daughter was born 2 months early. My husband loves my daughter so much and says no matter what I will always love her and protect her. Idk what else to do but play this sitting and waiting game with my ex until he files. I'm asking the court for full legal and full physical custody of her for her safety I don't trust him I have this feeling he would hurt her if he gets visitation rights. I just found out on January 27 husband and I are expecting baby number 2 in October however I am high risk pregnancy because I had my daughter early plus I had 1 stillborn set of twins and 1 set of twins whom I miscarried. I live in California please let me know what you think I can do. I don't need this stress of just waiting.