I have super sensitive and life-threatening allergies too and Disney has been GREAT! All of the food service locations have binders with lists of all the ingredients of particular items, managers or chefs will come out to answer your questions and will take your order on special allergy cards, and. At buffets or food courts, chefs will come out and make you a special plate of whatever items you want (that are safe) in the back to ensure no cross contamination. Most places have special allergy fryers for safe chicken strips and french fries. Many of the breakfast places have safe Mickey waffles and pancakes made on dedicated waffle irons and pans.
You're right, Tofutti and Rice Dream are off limits for anyone with sensitive dairy or nut allergies due to their cross contamination issues.

Same for the soy milks - Kikkoman and Silk aren't safe for me either for the same reason.
EPCOT was probably my favourite place to eat with food allergies because there were so many different options to choose from.
We had success everywhere we went, particularly the food court at The Land. They made me a special plate from the back and included, among the regular stuff, Enjoy Life chocolate bars and cookies. That certainly made my day! (I never knew Enjoy Life made chocolate bars before then.)
The Rose and Crown in the UK pavillion was also exceptional. The chef came to our table genuinely excited to make a special meal for us - whatever we wanted! We picked something off the menu which was safe (without the sides), but the chef took our order personally and delivered it to our table, and came out a few times during the meal to check on us and bring out safe side dishes. Best meal of the whole trip.
The bakery on Main Street (MK) had some cookies (pre-packaged!) that were safe, and we had those a few times. (We ordered Divvies cookies delivered to our hotel room, which provided us with plenty of snacks, but there's something special about buying and eating treats in the park like everyone else can.)
The chef at the POP Century food court was great at finding me options to eat there, and made them in the back for me. He then went out of his way to make sure the kitchen had fresh allergy-free items every day for me so I wouldn't have to wait for him to get them from other kitchens or thaw them from the freezer or whatever the case was. On that note, some chefs will do this if you ask them in advance - i.e. if you want Enjoy Life cookies available for dessert one night (or waffles for dinner) - call and ask for it a few days ahead of time (noted on your reservation) and if they have it on site, it will happen. They're pretty awesome that way!
Someone told me that pineapple Dole whip and the giant turkey legs are safe, but I haven't looked into it. Worth checking out! (They will have a binder with ingredients and may contains at the place that serves it.)
Hmm, what else stands out... There are a few quick service locations that are more allergy-friendly than others because they have dedicated fryers, special buns, and a few other allergy-friendly options available. The list is on the WDW Special Diets page. Restaurantosaurus was one of them, although their allergy menu is quite limited.
Basically, the best service and the most options were at sit down places. Food courts were next best. Always, always identify that you have allergies and speak to a manager (quick service) or chef (table service).
We did have one failure with food provided on a tour but thankfully I looked at my food before eating it and quickly realised something was wrong. (It's pretty scary to see your allergens on your plate!) Needless to say, Disney made major modifications to their tour food allergy procedures after that. I hear they are much better now for allergies in that department. Lesson learnt - whilst Disney gets five gold stars for accommodating food allergies, their staff are only human and mistakes happen. Keep your eye out and double check everything.
Have a safe and fun holiday!