What they are looking for is the booking name to match the birth certificate and photo ID name. If all three of these match then you'd be fine. If not, then you need to be able to show a paper trail of some sort to explain the name differences - such as a marriage license.
I get conflicting reports on whether or not the marriage license is always asked for, but it is just another piece of paper to carry and if the possibility existed that someone might raise a question I'd bring the marriage license along on the trip.
Another thing - do you have legal custody of the kids? If they are going on the cruise with you you may also be asked to prove that your ex-husband has no custody rights or if he does have rights that he has approved your leaving the country with the children.