I don't want him to show up and try to claim them or take them away. I would like to remove his legal rights.
Lawsuit / Dispute Attorney
If push comes to shove, the biological father can petition the court for a whole list of things such as custody, visitation, child support, preventing you from moving out of state (removal), et. al.,
Talk to a family law attorney before you do something that gets you hauled into court. What you want can be done, but it is too important to try and do this yourself and mess it up.