I would go to Publix! You're on vacation...that to me means NOT stepping foot in a Wal-Mart! I dislike Wal-Mart so much I haven't been in close to 4 years. (It helps that there are no Wal-Marts near my house, which is 5 minutes out of DC.)
When I worked at Disney, I always went to the Publix off of Vineland Road (I think that's the road?) It's back behind the Premium Outlets and the Dixie Stampede. It was always clean and reasonably priced.