I haven't taken a
DCL cruise yet (Mine is in Feb 2014!) so I can't tell you my first hand experience, but I have a son who is allergic to all tree nuts, so I've been reading lots of reviews about food allergies on this board.
Sounds like they go above and beyond to accommodate people with food allergies/restrictions.
I feel like getting to know your servers in the MDR and letting them know of your food allergies would be a safer bet than to just stop in quick bites as those places probably serve many people throughout the day so they won't remember if you have any food allergies or not.
Our allergy doctor suggested showing the Epipen to the servers whenever we dine out. Not to scare them but to let them know we're not some over reacting parents. By showing the Epipen, it really gets the message across, that if you ingest a tree nut, it's serious and we'll have to use the Epipen right there and then. It has worked, and so far, we haven't had to use Epipen on my. (knock on wood)
I tend to do this whenever we go to ethnic restaurants because some people are not as aware as typical American restaurants. Ethnic food tend to have lots of tree nuts in their food, too. I'm Japanese and food allergy is JUST starting to get awareness in the last decade.
Better be safe than sorry. Have a safe cruise!