I actually think there are quite a few non burger etc., cs places around WDW. The food courts at the moderate and value resorts are great. However, you may not want to figure that into your schedule! Some places where I have enjoyed some good cs are:
Epcot, Yakitori House, Tangerine Cafe, the places in China, Mexico, Norway, France and England all have many things, and burgers aren't on any of the menus! The Sunshine Seasons food court in the Land is also good, they have many options. Epcot in particular probably has the best selection of cs places that don't serve your usual theme park type food.
In MK, we like Columbia Harbor House, they have soup, sandwiches and of course the fish and chicken baskets. We also have enjoyed the wrap at Pecos Bills and Sosmic Rays has a rotissiere meal that can easily be shared.
In AK we enjoy Flame Tree BBQ and the sandwiches at Pizzafari.
In MGM, the ABC Commissary has some non standard items.
You can find fruit carts in all the parks which help balance out the diet. And many of the cs places do have salads which is nice.
And in DTD Wolfgang Pucks Express has a wide variety. There is also the Earl of Sandwich which is wonderful.
So I do think it can be done. And I can relate to your budget minded ways since I just cancelled all but one ADR for upcoming 10 day trip. Sometimes I just don't want to pay so much to eat, esp. since my kids are both now "adults". Just gets too expensive! I am kinda looking forward to it, we will definitely spend a lot less time eating in addition to a lot less money. Enjoy your trip!
Oh and another great way to save money is to share entrees at non buffet ts places. Most portions around WDW are large enough to share.