I am gluten-free and vegan, and I ate like a queen on the Dream. My daughter has food allergies too, so we were dealing with a lot. Here is my advice:
1) Get to know your head server, and become his or her best friend. Compliment him or her genuinely, ask where s/he is from, have conversations that are not about food. Your head server will take excellent care of you, especially if you can be a kind, polite, friendly sort of person (and I have no reason to believe that you wouldn't be one!). Your head server will be the one coordinating your dinners in the main dining room, and will often be available at the buffet during breakfast or lunch. Ask him or her where s/he will be for the next meal, and if possible, go there. If not, ask who you should talk to when you arrive at wherever you are eating.
2) Be pro-active. This really should be the #1 rule, but I'm too lazy to switch them. I have heard stories of people who were irate that their diet was not accommodated, but upon probing it sounded like they expected everyone on the ship to know who they were and what they could or could not eat. When you arrive at the buffet, tell someone who works there that you are gluten-free and you'd like some assistance finding foods you can eat. After a few days you will know what to do, or what to eat, but for the first few visits to the buffet ask someone, and if they can't help you they will find someone who can. Know that you might have to wait while they make your food elsewhere, and that it might mean your food is coming from the opposite end of the ship 10 floors down. Eating at the buffet might not really be a quick meal (or it might, but be prepared that you might have to wait).
3) Don't wait for them to ask you what you want to order for the next night's dinner, ask them if you can see the next night's menu so you can place your order.
4) At dinner, if you like something a lot, tell your server and head server. They will make sure it is available every night. I had gluten-free bread every night because they knew I liked it, and they brought out a spread my son really liked because he went nuts for it on our first night. The chef also made me this amazing dish that wasn't on any of the menus, because the head server suggested it and knew his friend the chef would knock it out of the park. Another day at the buffet a different chef came out and asked me what I liked to eat and created a meal for me that was great.
5) Pre-order your lunch for
Castaway Cay. While you can't take a meal off the ship, you can order your CC lunch at dinner the night before. They will tell you where to pick it up on the island.
6) You can only take packaged food off the ship, so if you are going on any excursion know that you might not be able to eat much at your destination, and plan accordingly. Bring a granola bar or something safe.
7) Again, don't assume that everyone on board knows about your needs, and you will be fine. Just politely let people know your needs, and they will gladly help you out. I pretty much behaved as I would at any other restaurant, except that the
Disney Cruise Line food service people were WAY more knowledgeable about allergens and ingredients than most people I have encountered elsewhere.