We just got back, and were pleasantly surprised at how easy it was to eat healthy while there. DH is on a medically-restricted very low fat diet, so we were a bit worried about eating in the park. As it was we ate almost all our lunches and dinners there, and had no trouble at all. We ate turkey or chicken sandwiches, fruit plates, side salads, big salads, chicken salads, baked chicken, barbeque chicken, grilled chicken, pasta with marinara.
DH was very happy that he wasn't stuck eating greens all day long. It was a lot of chicken, but because of the variety in preparations, it didn't get old.
Edited to add: There are only 3 full-service restaurants at DL. One of them is Carnation Cafe on Main St. I believe Blue Bayou is another, but I can't remember the third... So unless you are willing to only go to these three, you'll be doing some counter service.