I've always tried to eat relatively healthy at the World, and have tried to eat gluten/dairy free on numerous trips, here's what I suggest for counter service:
Magic Kingdom:
Cosmic Rays-> Turkey Sandwich w/apple slices (instead of fries, it's multigrain bread, turkey, tomato, pesto, provolone cheese, red onion. It's delicious!!)
Columbia Harbor House-> Salmon filet w/veg or rice or whichever
They also have a delicious chicken salad
Tortuga Tavern-> Chicken or Vegetarian Burrito-> chicken, beans, a bit of rice, and you can pick your own toppings. Comes in a tortilla
Epcot:
Sunshine seasons->Variety of fruit & cheese plates, fruit plates, healthy chicken & vegetables,etc.
Morocco Pavilion-> Has some nice slider samplers with felafel, lamb & chicken inside them and various sides
Japan-> Any kind of grilled chicken with rice, or sushi that isn't rolls is awesome for you, or fish & rice/veg
Mexico-> I'm sure has some kind of buritto/fajita stuff that isn't deep fried!
DHS
Starring Rolls-> Sandwiches, turkey or ham with veggies. Though there is a GIANT amount of white bread with them, I usually take off a slice, or rip it in half.
Down near ToT-> Plenty of places you can get a grilled sandwich or salad
AK:
-I would suggest Flame Tree BBQ, as they have lots of grilled chicken and veggie options!
In terms of breakfasts, I like to just get say, a bagel and a side of fruit, or a side of eggs, build your own omelet is great, (even with the yolks, they are good for you!) I will also order the "bounty platter" sometimes, and eat the eggs, a bit of the protein portions (as they're usually greasy) and one of the carb portions and the potatoes
Sandwiches are readily available and delicious all around Disney, but I find sometimes there's just too much bread on them so I end up taking off the top slice and eating it as an "open face" sandwich!
Also look at some of the kids meals for better options, I know they've instigated the "Mickey Check" at some places (IE, Be Our Guest) on the kids meal I got a grilled turkey sandwich on multigrain bread with yam wedge "fries" and some apple sauce. Kids meals are also portioned a lot better, they're much closer to what you should actually be eating, bahaha! Other places you can get say, a little sandwich with apple/carrot slices & grapes!
As for sit down places; I try to sub out anything that has a lot of dairy (thanks to lactose intolerance, booo) as in, roasted or baked potato instead of mashed, no cream sauces, not potatoes at all, just extra veg with the protein.
Also remember that everything is okay for you in moderation, I usually "allow myself one treat" per day that I'm there, whether that's half a red velvet cupcake, or a coke because I need the caffeine, it helps!
Hope this helps & have a magical trip!