I noticed on my July trip that a lot of places are offering different types of sandwiches and healthier choices, so be sure to read the entire menu...or even better, ask a cast member! There are lots of choices besides burgers and chicken nuggets.
I think Earl of Sandwich @ Downtown Disney has a great selection.
If you like Mexican, Tortuga Tavern (open seasonally and hours 11am-3pm) is usually a hit, as well as Columbia Harbor House (some seafood but sandwiches & other things, too) @ MK.
Epcot's World Showcase, of course, is fully of a variety of food, but there's also Sunshine Seasons in the Land pavilion.
My mom recently had a panini that she loved at the Studio Catering Company @ HS.
I've heard rave reviews about the Flame Tree Barbecue & a friend of mine cannot leave AK without a hot Italian sandwich from Pizzafari.
Hope this helps! Enjoy!