Any registered voter who changes his or her name by marriage or otherwise, shall be required to register anew and authorize the cancellation of the previous registration; provided, however, that if the change of name takes place within a period during which such new registration cannot be made, next preceding any election or primary, the elector may, if otherwise qualified, vote upon making the following affidavit before the judges of election:
     “I do solemnly swear that I am the same person now registered in the …. precinct of the …. ward, under the name of …. and that I still reside in said precinct.

    If the voter whose name has changed still resides in the same precinct, the voter may vote after making the affidavit at the polling place regardless of when the change of name occurred. In that event, the affidavit shall not state that the voter is required to register; the affidavit shall be treated by the election authority as authorization to cancel the registration under the former name, and the election authority shall register the voter under his or her current name.