Can I Ask My Ex To Sign Over His Parental Rights?

I have been wanting my ex to sign over his parental rights since my son was born. Lawyers told me that I couldn't do it unless I had someone else to adopt him. My son is about to be 11 and doesn't know his father at all. My ex moved 3k miles away when my son was less than 1. He was abusive to me and after an altercation at his mom's wedding when my son was 3, he locked us in a hotel room for hours and then said he couldn't be in our lives and disappeared. He also was behind on child support for half of my son's life. The problem is that my ex randomly sold his house, paid back most of his support, and moved back to the area when my son was 8 and demanded to see him. I told him to take me to court to get something official written up. It took him 2 years to do so. He also made it take another year because he kept not showing up to court and not doing what the court asked. I told the courts I didn't want him in my son's life, as he was abusive and my son didn't even know him. That obviously didn't go over very well. Two months ago they gave us a court order that says he has to find a therapist and attend therapy with my son and then afterward he would be entitled to virtual visits 3x a week.

Here's my question. I still want him to sign over his rights. I am married and pregnant with twins and my husband is in the military. My son sees my husband as the only father figure he has ever had and wants to adopt him. It would make life a lot easier to not have to petition the courts to ask permission to move every time my husband gets new military orders, my husband would have legal rights to my son that he currently doesn't have, and I wouldn't have to worry about my ex's flippant moods where he decides he wants to be a dad and then causes chaos until he leaves again.

Can I ask a different lawyer to draw up paperwork to have him sign over his rights so close to the other court order being established? I told my lawyer that I wanted him to sign over his rights but I can't even really confirm that he ever mentioned it to him. I live in NY which makes this more difficult because the laws benefit deadbeat parents.