if you want to assign guardianship of the child to a family member, you will still need to inform dad, have him served. if he contests to it, barring any legal reason he is unfit, he will be granted custody.
if he is found unfit by the state laws, then it may be possible to have it court...
most likely, the records aren't missing. they are expunged.
and no, you will not be gaining custody of the child based on mom having a past history of DUI.