Can my sons father willingly sign away his rights?

Right now, since the custody order is out of Ohio, Ohio has jurisdiction. However, since it seems (from the explanation) that you are here in Indiana (and, more importantly, your child), you can attempt to have Indiana establish jurisdiction by having the custody transferred to Indiana. In Indiana at least (I do not practice in Ohio), you can seek to have your current husband adopt your son if the biological father consents. (There are some exceptions, but I am speaking in general terms). If biological father consents, the arrears would fall away after the finalization of the adoption. Yes, you should go through a formal adoption process for many reasons. You should retain an experienced custody lawyer to represent your interests. Best wishes to you.