Mother knows best in Missouri

I have a 3 yr old & her fathers name was put on her birth certificate 1 year ago - I was a single mom & was forced to put his name on the b.c. if I wanted to continue to receive medical support from the state. He has been ordered to pay child support, name is on the b.c., he has not had any contact with us for 1 1/2 yrs & has not seen her for 2 years - nor has he ever paid a court ordered child support. I am now married ( got married 3 months ago) & want the bio-fathers name off the birth certificate - but am not ready for my new husband to adopt my child.
Child supprot has sent the bio-father many letters in attempt to collect - he doesn't hold down a job & is constantly moving around - no one knows for sure where he is.
This is typical of this man - found out he has a 15 year old son who he has done the same thing to & had even kidnapped him when he was 2 yrs old & took him out of state. His son's mother did file a report but it was in a small town & there are no records of this anymore. The bio-father has been a known drug & alcohol addict, manic depressant, run in with the law - the list goes on & on.
I have contacted an attorney & this particular attorney only suggested that I have my new husband adopt my child & prove the bio-father to be unfir, abadenment,etc - this is timely & pretty expensive. Do you have any other advice?
My sole concern is that if something were to happen to me, that she would be handed over to her bio-father that she doesn't know, is unfit to raise her & so on...... I believe mother knows best on this but how do I go about getting my name soley on the b.c?

Concerned in Missouri!