My question is... as a mother with sole legal and physical custody of my child... can I force the biological father to sign away his parental rights if he refuses?

No, you can't. He would need to completely abandon the child and be unreachable for some period to get an involuntary TPR through the courts.

My question is...as a mother with sole legal and physical custody of my child....can I force the biological father to sign away his parental rights if he refuses?
I have been through this too!!! If he has abandoned the child, child support has a hard time to find him and get him to pay, and you have proof of all of this, you should be able to get him to sign off rights. If there is a step-parent wanting to adopt the child it will make it a bit easier. The laws vary from state to state. Your best bet is to do a google search on "step parent adoption" this will give you many web sites on the laws of adoption by state and the information needed to pursue the father to sign off his rights to the child. Hope this helps, you can always email me with other concerns and I will try to help you as much as possible.