Samsma - your ex sounds delightful. Congrats on your divorce.

Yeesh.
If you have FULL LEGAL custody of your child, your ex should not be able to stop you from taking him on vacation or out of the country, and you don't need any kind of "authorization" from him unless your final divorce decree so states. Now if you are causing him to miss his visitation, then that's another negotiation all together, and I wish I had some advice for you, but some people have to make everything difficult and I am sorry you have to go thru that. If you are concerned, you should call your lawyer and ask if you need any kind of papers with you; I doubt it but it cannot hurt to ask. If s/he is a good lawyer (and I sure hope so), s/he would answer a simple question like that gratis. If you want to take your final decree, it couldn't hurt, but you should not need any kind of notarized statement from your ex. Sadly, no one can stop him from saying stupid stuff to your kids or making threats, but legally, in general, he would have no right to stop you from going on vacation any more than he can stop you from taking them to the grocery store. Further, if your DS need medical attention anywhere, not just on the cruise, you obviously have the authority to seek that w/o your ex. Here in the US you would not have to "prove" custody (in fact, who would know if you were divorced or married, as someone else pointed out?!) One parent alone can authorize medical treatment.
If I were you,, though, I would get passports for everyone. I know it is an added expense, but they are good for 10 yrs and nowadays, I would not leave the country w/o one. It is absolute proof of your citizenship and heaven forbid, should you ever require assistance from US Consul, it is very helpful. As for the last name thing, nobody is going to notice or care, frankly. My last name is not the same as DH or DD's and nobody blinked. I showed our passports on embarkation, then they went into the safe until debarkation. If you take DS's BC, they will see your "old" name on it and if necessary you will tell them that you've been remarried since then.
Does this help you at all? I hope so!! Please don't worry too much and have a wonderful cruise. Don't let your ex ruin your fun.
Allison