How can my new husband adopt my children, his step-children?

To accomplish an adoption of stepchildren, your husband must file suit and give notice to the father of the child. It is possible that the father would be agreeable, and if he is, he can sign an affidavit to that effect. The judge may decide that an attorney should be appointed to represent the child (attorney ad litem) in a step-parent adoption circumstance. If so, you can expect to pay that attorney as well.