When can someone file for abandonment of a child in the state of Florida?

My husband has been in my son's life since he was 3 months old but my son's father only sees him on his birthdays and christmas (which are a month apart). He stopped forwarding his address in the mail system two years ago and started working under the table when I signed up to receive child support. He has not taken a paternity test yet but he did sign the birth certificate. I'm worried about what will happen because my husband is about to join the navy and we're about to out of the state of Florida. I do not want him to have any rights to my son at all, it also worries me for the fact that his now girlfriend does not have custody of her children (her ex-husband received full custody in the divorce). Any help?