There is no easy answer for this - every situation will be different.
The passport question is a good point since few people go ashore with their passports (and we don't advise doing so - no one plans to miss the ship). So, it can be tricky traveling in the Caribbean without a passport, and even worse if you have to fly back to the U.S. since passports are required for flyers to enter.
Now, there are such things as American Embassies who handle problems like these. They can get you a temporary replacement passport. But it takes time and that = money.
There are also restrictions on trying to join a ship in a different U.S. than the one from which it originally sailed. Something about the Jones Act (although the logic of that escapes me right now).
In any case - it is never a pretty picture, but you won't be forced to retire in Anguilla just because you miss the ship - you will get home. It will just cost you more than you thought.
And I'll bet travel insurance doesn't cover that,either.