We've been going to Disney for years in part because of how they handle food allergies.
You can email Special Diets at WDW to give them your allergy lists and they will send you a list of safe foods from carts, etc. (ie popcorn, pretzels, ice cream, etc.)
At Table Service restaurants, make sure you note it when you are making the reservation. There will be a place where you can check which allergies apply to your party. When you check in, they should mention the allergy. If they don't, you should mention it then, before you sit down so they can notify the chef to come talk with your table. We've been scolded (yes, really

) a few times for not mentioning it then before we sat down (but few and far between).
Counter Service restaurants, they have a manager come over and take the order. They show you a binder with the ingredients most of the time so you can double check. They put the order in specially to avoid contamination.
Be prepared to put your patient hat on, sometimes this process takes a while (mostly by having to wait until a manager can come over to take the order.)
It may be a process, but Disney takes food allergies very seriously, so it's worth it. My DD has a severe peanut and tree nut allergy and has thankfully never had any type of reaction, and we go several times a year.
A lot of people says they will make special items for you~ that used to be more prevalent than it is now. It seems like they have plenty of items to accommodate different allergies now, so that special treatment has become more scarce now than it used to be. This is just our experience. Most of the alternative items for peanut and tree nut allergies seems to be the gluten free options. Gluten-free bread, cookies, desserts, etc.
My tip is to be consistently vigilent. Just because a burger bun is ok in one place doesn't mean it's ok in another. Same with chicken, pizza, ice cream, etc.
If you are doing Counter Service on the Dining Plan, you can substitute out a bottled water for a dessert since many of them are not ok.
Hope that helps a little~ plenty of safe options! Have fun~