The birth certificate and the picture drivers license should do just fine. If you think you will be taking more cruises I'd suggest thinking about just getting a passport.
It just seem so much easier than taking all those papers. We got them a few years ago and its just much easier and convient and they are good for 10 years. Here is a link to help you to get one if you want to:
If you've just recently married and your new name does not match the name on your birth certificate and drivers license, you should definitely take your marriage license! Especially if your airline tickets and cruise documents are in your married name. Why risk it?
What you have is perfectly fine. I might suggest that you invest in getting a passport though as you will most likely take more cruises and not all of them will be to the Caribbean. They are just so much easier to deal with.