There is a brand-new Winn Dixie where we shopped, I think even closer than Publix. You have to head out past Downtown Disney and through the Downtown Disney Hotel area. At the stoplight, where you would head straight to enter the Crossings area (home of the overpriced Goodings), make a left. Winn-Dixie is up just a mile, in a new plaza on the right Also up this way on the left was a nice, big Walgreen's (we needed zinc oxide for my pale daughter!)
The Winn Dixie is nice because it is new and the prices were reasonable. I got one of their discount shopper cards to use twice a year when I am in Florida. They even had an area with Florida souvenirs.
Have fun!