To expand on the above directions... I love using
http://www.mapquest.com/directions/ to help me plan my excursions when I'm going to Orlando.
This Winn Dixie is located at
11957 South Apopka Vineland Rd
Zip Code is 32836
The Walt Disney World Swan Hotel is located at
1500 Epcot Resorts Blvd
Zip Code is 32830
(sorry, I don't have the Beach Clubs address handy)
Plug in those addresses and it says it's less than 6 miles to the Winn Dixie from the Swan.
Basically, from the Epcot area, take E. Buena Vista Drive toward Downtown Disney, it's under 4 miles away. You'll pass DD's Westside, Pleasure Island and finally the Marketplace area. When you get to Hotel Plaza Blvd, take a right. It's less than a mile from there to 535. (At this point, if you were to go directly across 535, you'd be at the Crossroads and see the Goodings supermarket on the left - This is also where you are leaving WDW property) To get to the Winn Dixie, turn left on 535 which will basically turn into South Apopka Vineland Rd. The Winn Dixie is less than a mile from Hotel Plaza Blvd on the right hand side of the road.
If I remember correctly, there is a Walgreens in the same area. You'll also see lots of restaurants along 535, including Red Lobster, fast food places, IHOP, Perkins, Jungle Jims, etc.