I’m confused about this situation.
Even if NCL allows you to sail with a birth certificate, how do you plan on getting home from repositioning cruise? Any foreign airport or cruise terminal on new check in will require a passport from US citizen.
Also being on so many previous cruises you should have a passport by now.
Because traveling soon, you should be able to get an emergency passport. This will only work if foreign country allows passport issued less than 6 months ago.