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.
59 Carnival: Mardi Gras, Ecstasy X2, Fascination X6, Celebration, Elation X5, Glory X5, Fantasy X4, Valor, Holiday X3, Imagination, Liberty, Spirit X2, Freedom, Destiny X2, Splendor X2, Dream X2, Conquest, Legend X3, Victory, Sensation, Paradise, Inspiration, Magic X3, Triumph X2, Miracle X2, Breeze, Sunshine X3, Pride
7 Norwegian: Sky, Jewel, Dawn, Epic, Pearl, Breakaway, Getaway
8 Windjammer: S/V Legacy X3, S/V Polynesia, S/V Caledonia, S/V Diamant, S/V Sagitta X2