Your passport will be checked by the cruiseline (not stamped) when you get on the ship. It normally (there are exceptions) is not checked or shown again while on the cruise. Where you might have trouble is re-entry into the United States.
US passports are scanned upon re-entry. Don't know what would happen with a NZ passport.
8 Norwegian: Sky, Jewel, Dawn, Epic, Pearl, Breakaway, Getaway, Escape
10 Windjammer: S/V Legacy X3, S/V Polynesia, S/V Caledonia, S/V Diamant, S/V Sagitta X2 S/V Mandalay, S/V Vela