You REALLY need to contact Special Diets NOW. The more information you're armed with ahead of time, the better off you are. My girls have complex lists of food allergies as well yet we do really well at WDW. DD9 can't have milk, eggs, peanuts, hazelnut, brazilnuts, citrus, cranberries, blueberries or sesame. DD12 can't have gluten, milk, eggs, yeast, citrus, MSG or peanuts.
When Special Diets sends me all their info on various allergies, I come up with a chart of what safe foods I can find where and I print it off, laminate it and carry it around with me in the parks so I know where I can find things. Here's the one I put together for our upcoming August trip. You really do need to do this every time you go to WDW though. Things can change. I was surprised that I had to make changes from my chart from my June trip, but there had been a few changes when I recieved my new lists for August when I returned. Note that this chart won't work for you because I haven't verified YOUR entire list; it's what I've customized for my family. Each individual with food allergies would need to come up with his/her own personlized version. Knowing what can be found where is also helpful because many times CMs don't even know that their location carries certain allergy foods until I assure then that Special Diets told me the products are there and I ask them to look again (they always find the stuff and are shocked because they had no idea).
I've also found that the chefs in the food courts at resorts have a lot more flexibility than what you'll find at CS locations in parks regarding the ablity to make a wide variety of safe meals. At parks, you can order only based on the limitted menu there. At resorts, they can custom make any kind of food that you want as long as they have the ingredients.
I'm not sure on the strawberry but I believe the Itzkadoozie popsicles that are found all over the parks are free of all your other allergens. I know that they do have some real fruit which is why my kids can't have them (they have orange and lemon juice in them) but maybe there's no strawberry. Ask Special Diets about them. They're easy to find all over the place and would be a great snack for boosting blood sugar and cooling you down.
There are lots of places around the parks where you can find fresh fruit, both cut and whole. These would be perfect snacks.
I would suggest that you do carry some snacks with you. You'll have a stroller anyway (I'm assuming with a baby young enough to still be nursing) so it'll be easy to store snacks in the basket. Bring a soft sided cooler to keep anything temperature sensitive cool. I'm suggesting bring something in case your blood sugar start dropping. You don't want to have to start searching for a place to find safe snacks (though having a chart like the one above at least means you'll know exactly which places to look for rather than having to ask for help finding something and then having to search through a binder).
Those Enjoy Life chocolate chip cookies are AMAZING!!! We also use them at home. The cheapest place I've found them is
amazon.com. I'm set up with their subscribe & save to get an additional 15% off and they just deliver them to my house on a regular schedule.
TS locations for the most part are going to be easier than CS, especially at parks. At your own resort, CS can be just as easy.
Do prepare for the fact that meals are going to take longer for you than for most people because your food has to be specially prepared rather than the mass production done for everybody else.
Is Rice Dream ice cream safe for you? It's served at all sorts of locations around WDW. Some places will also carry rice milk.
I do place an order with Garden Grocer for some breakfast foods so we can eat in our room. Many resorts will have rice milk in the kitchen that you can request but for me it's just so much easier to have it in your room (plus I can ensure it's the type my daughter likes; she's EXTREMELY fussy because she has some obsessive compulsive tendencies). It's just so easy to have breakfast in your room rather than have to go over to the food court to have to wait for a special meal to be cooked.