I also make a point to eat healthy and my definition of healthy is the same as yours - unprocessed, whole foods - lots of veggies, fruits, whole grains, lean proteins. I limit deep fried, but otherwise do not restrict fat per se, and if I feel like eating some fried calamari, I go ahead and do it. Lunch or dinner at TS restaurants were easy. The Wave, Coral Reef, Kouzzina, Cali Grill, HBD, Sanaa were all healthy. I had fish as my entree at all of those and they all came with veggies. I also had the whole fried fish at Narccoossee and it was very tasty, definitely worth going for. Surprisingly Tutto Italia was also a good choice. The antipasto platter included artichokes, olives, fresh mozzarella, cured meats, cannellini beans. I had a salmon as my main entree with olives and artichokes.
Asian-inspired places were a a little tougher to navigate. Salmon at Yak and Yeti was good- it wasn't fried, it was served with nice veggies and brown rice. The tuna appie was good - seared and served with an asian slaw, but the green beans were totally deep fried. Ohana wasn't particularly healthy. The potstickers looked very processed (did not look homemade, but I could be wrong), the dressing for the salad and the dipping sauces seemed to me like they would have quite a bit of sugar and sodium. Noodles did not look particularly nutritious. My noodles at Kona seemed healthier - at least they had shrimp and came loaded with veggies.
50's PT wasn't particularly healthy for main entrees. I had the squash agnolotti and it was OK, but the seafood salad appie was great - very fresh and lean. It's comfort food and given what it is, it was fine. Another comfort food place - LTT was also just OK. I had the veggie pot pie. The veggie filling was OK. DH had the pot roast. It actually looked pretty good, albeit a huge portion.
At breakfast I stayed away from things like bagels (no 100% whole grain option anywhere in WDW), but breakfast buffets usually had an omelete option, fresh fruit, and ham. Some even had some veggies. Tusker House even had quinoa.
Snacks were pretty tough - too many refined carb options. As far as I could tell, nothing baked in WDW was truly whole grain. Fruit cups were easily found and they were the best snack. I hear there are carts that sell nuts, but I never came across one when we were looking for a snack. We had lots of cappuccinos as a snack, water, and yes, even some cashew patties and ice cream.
If you are trying to eat healthy, you definitely can in WDW!