In the industry, phrases like "notary witness closing" use the word "witness" loosely. When the notary takes an acknowledgement, the notary notary asks "did you execute this instrument as your free act and deed""* and witnesses the signer say "yes".
When the notary administers an oath for a signed document, the notary asks "are the contents of this document true to the best of your knowledge, so help you God?"* and witnesses two things, first that the signer signed, and second that the signer said "yes" to the oath.
So chances are the notary won't have to sign on any witness lines (unless that's customary in your state for certain documents).
*or words to that effect.