Can you share what some/all of your allergies are? There are a lot of DISers here with allergies or with family members who have allergies. We have a lot of experience to share about where to eat in the parks or in the area around the parks.
I also have multiple allergies and sensitivities, but have not been able to rely on getting ingredient lists from the parks before a visit. Disney is correct that suppliers can change at any time, therefore ingredients can change at any time. It does no good to rely on an outdated ingredient list. Your best bet may be to ask a chef to speak with you at each location where you dine, CS or TS. For TS, note on every reservation that you have allergies, what they are, and that you would like a chef to speak with you. Repeat this at check in and with your server. It helps if you have your allergies and safe foods printed on cards that can be given to the chefs so they know what they're working with. Even at CS locations, a chef can walk you through the menu and possibly adapt something for you. Always have your meds handy and some safe foods with you to keep you going. The combo of bringing my own food/snacks and speaking with the chefs has worked fine for me so far at DLR. By the way, if you can time your meals and reservations for off peak hours, that will really be to your advantage. The chefs will have more time for you if the kitchen isn't swamped. And since making something custom for you can take extra time at CS locations, going when the kitchen isn't super busy already will be helpful.