You will need to select your CS restaurants based upon the kids meal menus. I've been doing that for years because I don't feed my kids chicken strips or french fries and I don't think they've had a hotdog in at least a year.
Epcot will be a great place for kids CS: we like the teriyaki chicken at the Yakitori House (I eat it too). Cosmic Rays at the MK has a chicken salad (chicken strips on dark leafy lettuce). My kids usually eat the chicken at Tusker House and we rarely eat anything at MGM.
CS menus have been changing a lot, so don't expect the same thing when you go as you're seeing right now.
For TS you will do best at buffets or at Signature Dining like Artist Point. When you call in the dining ask them to note the food allergies on the screen (there's a dropdown box for them to check). Disney does a great job allergies and dining.