ClanMc has great suggestions, that I would echo. I would look at all of the menus (we usually use allears.net) and find places that have dishes that sound good to you, or sound like they would be modifiable. If there is a particular restaurant you are unsure about, you can search that with the name of the allergy - "Ohana dairy" - in this forum, and will likely find what you are looking for.
We just visited over Christmas week, and three of the four of us have allergies. Mine are mushroom/mango, and I am a vegetarian. Friend #1 is allergic to strawberries, peanut, tree nut, shellfish, fish, eggs in large quantities (baked into a cake is ok,) soy in large quantities (lecithin is ok,) limited milk, limited wheat. Friend #2 is allergic to latex (her dishware needs to be rewashed,) banana, kiwi, avocado, chestnut, hazlenut, cashew, mango, passionfruit, papaya and guava. We all ate very, very well.
Of note, we went to Chef Mickey's for breakfast (we thought it was brunch *sigh*) and the chef walked the buffet with us. The fruits were off-limits, obviously, and there were some fruit compotes. IIRC, there were no nuts at all. The wheat allergy is not severe, so she did not mention it to the chef and just avoided obvious gluten, but there did appear to be a good selection of GF items, so long as the breads table was avoided. Dairy might have been a bit harder. If you are concerned about cross-contamination, they will bring out fresh food from the back (for example, friend #1 wanted watermelon, but it was next to the strawberries, so the chef just brought her out a fresh plate.) She might have asked for one other item, but mostly ate off of the buffet, since there were so few problematic foods.
We did not do any princess meals.
Our favorites for allergies are: Plaza (I am pretty sure poor Chef G is frightened of us, but he is awesome) and Le Cellier - although they have changed head chefs, so our favorite favorite chef is not there any more, but the new chef handled us just fine.
'Ohana has been really hit and miss for us; we went in July '11 (friend 1 was not along) and it was fine. We went in September '11 with all three of us, and it was the single worst Disney dining experience we've had. We tried it again, in August '13, again without Friend 1, and it was amazing. We had Chef Susan and I have every confidence she could have fed Friend 1 with ease. Garden Grill was great for us, but we have read mixed reviews. Via Napoli was our surprise on this last trip, our chef was amazing and the pizza was sooooo good! And we did Flying Fish in August, which is a sig so maybe not where you want to take the grandkids, but holyyyy cow was it great, no issues with allergies.
Be Our Guest has handled us well - they seem to rotate chefs (as opposed to some of the other restaurants, where we seem to always encounter the same chef each visit) but we've never had a problem. We had Chef Jerry this last trip and loved him, Chef Chappy before, plus various other chefs have handled us when we've gone for CS lunch.
CS, you might check out Sunshine Seasons. The head chef - Chef Paul - from LC is over there now (or so our server at LC told us) and that guy gets allergies like no other. We walked into LC in July of '11 with Friend 2's newly developing latex allergy, and he actually created a protocol with us that we were able to explain to each chef as we went along, and is now the protocol that special diets sends out for latex allergy. We had some time to speak with him on a later visit and he was able to talk to us about some resources to pass on to local restaurants about how to set up an allergy-safe kitchen. So I am guessing Sunshine Seasons is up to snuff with the allergy items, although when we went in December, we were all able to find safe items without having to call for the chef. The only other places we have invoked the allergy system at CS are Pecos Bill's (they're always great) and Pizzafari (where I was able to get a dairy-free pizza, just had to wait a few minutes. I know they can do gluten free too.) Oh, and Casey's, the guy managed to upsell me to vegetarian nachos when I was waiting in line with a friend, not intending to get anything at all lol. I know they have GF buns. But with a dairy allergy, it might not be the best fit, not sure about their hot dogs and they have lots of cheese both in the kitchen and the toppings bar.
We have eaten at a good number of TSes on property, so if you are thinking of a specific one maybe someone here can give you specific advice
