I avoid restaurants whenever possible too, only trusting two local places and those at Disney parks. All I can say is bring chef cards listing your allergens and their derivatives, ask and verify everything when you order and again when it arrives. Avoid anything not on your specific plate - like the bread bowl on the table. I avoid desserts too, sticking to the fruit plate or bowl of berries. (Disney Cruise had Enjoy Life cookies! I ordered them every day.) Pre-ordering is highly recommended too - ask for it if they don't offer it. If anything or anyone doesn't meet your standards talk to the maitre d right away or guest services before you eat. We had to do that on one cruise and they were able to help. Your server should be transparent with everything and be able to answer all your questions or get the chef to answer them for you. (i.e. I asked what type of soy milk they were using and if it was nut-free - it was a brand I knew that used shared lines, so I requested that they not use it in any of my food and they added a note to my file without hesitation.) The more you ask questions and the more they get to know you, the easier it gets.
That brings me to another thing we had trouble with one day on another cruise line (Princess) - avoid "any time" dining. They have it as an option on Royal Caribbean too. We were forced to do that one day, everyone was, because of some delay at one of the ports. It was exhausting explaining allergies all over again to a different server team for that one day. If you stick to traditional/standard dining, where they assign you a time and table that stays the same every night, it is MUCH better. That way, you only have to deal with one set of servers and once they know your allergies and you are confident that they understand, you can relax a little (but still be hyper vigilant). You'll still have to explain things at breakfast and lunch, but even then you can find a few of the same servers to deal with every day.
Definitely stick to the main dining room whenever possible and ask your head server or the maitre d for their suggestions on eating elsewhere on the ship. They were the ones who told me to ask for a chef at the quick service places, and which ones may have safe options (i.e french fries made in a dedicated fryer).
Shortly after boarding on the first day, we went to guest services or wherever it said to change dining options to make sure allergies were flagged on our file and that our servers were all trained in allergy ordering. That may be a wise thing to do as well. I am not sure if they can check this from shore (at check in or over the phone), but it's worth asking or at least noting on your reservation ahead of time too. If your server isn't there at breakfast or lunch, be sure to ask for a server with allergy training for those meals.
Oh, what else... stay away from room service. On every cruise we have been on (Royal Caribbean, Princess, and Disney) they have told us that they couldn't guarantee allergy-free food from room service. Instead, we ordered some safe snacks at the dining room to take back to our room. (One chef made me a huge plate of safe cookies to bring back to my room, we found whole fruits at the buffet, which we washed before eating, and we found a few other little things as well.)
Cruises can be stressful for allergies, much like any vacation, and I bring a lot of my own food with me - prepackaged snacks mostly. Most cruises will have a great chef who will work with you and make incredible things for you. (I'll never forget an elegant dessert one Disney Cruise chef made for my husband and I!) Others may limit you to the bare basics like steamed rice and vegetables. We've had both experiences with Royal Caribbean. Needless to say, I never end up gaining weight on cruises like most people do... We go on a cruise every two years or so, two of which have been with Royal Caribbean, and we don't have anything bad to say about them. Some days were frustrating trying to find safe food that I wanted to eat or having to wait an hour for them to prepare a safe meal or snack, but most of all I remember the fun activities we did and places we explored. Eating on the cruise ship was less stressful than trying to visit all of those ports on our own and trying to find safe places to eat in each place.
I hope you have a great vacation.