Can my new boyfriend adopt my son even if we are not married ?

It will depend on where you live, if in the US some states are more liberal than others about this, but you would have to show the court a history of living together in a relationship. Most states are allowing adoption for life patners even if they are not, or can not get married.

First of course before you can even consider that, the bio-father will have to sign over all rights and sign permission for the child to be adopted,

In Ohio you have to be married a year before the husband can adopt the child... you would have to check the laws where you live.

That, and your child's father would have to willingly sign over his parental rights in order for the child to be adopted.

Possibly, The bio father's consent is required. But the judge would certainly prefer that you were married.