In Minnesota, as defined by section 245A.02, In the case of a stepparent adoption, a background study must be completed on the stepparent and any children as required under subdivision 3, paragraph (b). In other states, stepparent adoptions are exempt from background checks. In essence, the state puts faith in the custodial parent that they are making the right choice for their kid.
So the court IS going to look into your criminal past. It will be a matter of whether or not the Court feels your crime should stop you from adopting or not.
If father objects to your adopting, the road will get more difficult obviously.