Ok! So with "winging it" are you going to try to get walk-ups at any table service restaurants, or stick to quick service?
At most restaurants now (both TS and QS), they have allergy menus. They're kind of bass ackwards in that the items that state the allergy are the ones that are safe for you. For example, if it says "For Gluten, Egg, Peanut Allergies", it means it's safe to eat.
I also recommend buying the "Gluten Free Disney Dining" phone app. It's $2 or $3 in the app store, but it's incredibly detailed and identifies the safe items at every snack cart and restaurant.
At Magic Kingdom, Columbia Harbour House is really good. They have really delicious chicken tenders, and house-made honey mustard sauce. They have other entrees too, but I've only had the tenders because they're awesome haha
If you can get a reservation for Be Our Guest at lunch, it's one of the best meals I've ever had at Disney. Their turkey sandwich and potato leek soup are both amazing, and they also have yummy GF cream puffs (although they are sweetened with a fake sweetener, so they have that slightly funky aftertaste)
I haven't eaten at Cosmic Ray's, but they have some more standard faire of hot dogs, burgers, mashed potatoes, rotisserie chicken, etc.
For snacks, the popcorn, turkey legs, and Mickey ice cream bars are all GF.
Very quickly, I also recommend Sunshine Seasons at Epcot, and Pizzafari at Animal Kingdom. If you want to delve into Table Service and look for some reservations, it'll definitely open up more options! I don't have a lot of time to get into more general overviews of the other parks, but feel free to send me a private message with any of your loose plans, and I can give some recommendations!