Your husband does not have to tolerate someone who "harrasses and berates him nearly daily". Being the mother of his child does not give the woman any special right to do so.
If this were in WA, he can petition a court to restrain the woman from continuing to contact him. However, since the woman appears to be in OR, your husband likely will need to file the petition in OR. He would need to contact an attorney in OR to see whether OR would restrain a woman who "harrasses and berates him nearly daily".
Without a child being adopted, your husband cannot voluntarily terminate his parental rights and responsibilities. However, as the courts realize, there is not any practical way that the courts can force a noncustodial parent to be a parent: keeping in contact with the child, provide moral support to the child, and all the many things parents do for their children.
A better solution, if your husband wants to be a part of the child's life, is to get the appropriate court orders to restrain the mother from interfering with his relationship with the child.
Your husband should review his specific facts with his attorney to find out his legal options.
I assume your husband is on the child's birth certificate and on the paternity affidavit? The answer to that question may influence my advice. Is there a reason the mother's significant other doesn't want to adopt the child, if that is the case? Is it because they're not married so he's not really a step-dad? I don't believe the Court would allow the bio dad to relinquish his rights without another individual stepping in his shoes (i.e., step-parent adoption). In a typical, successful, stepparent adoption, the biological parent's (the one who relinquished) financial obligations cease accumulating though they're still responsible for any back owed child support that accumulated to the point of relinquishment. I suppose the bio parent could continue paying even beyond that point to the mother directly...but I have never seen that happen.
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