The easiest to get to is probably the Publix on West Hwy 192. It's probably about the same distance from SOG as the Winn Dixie on SR 535, but it's easier to get there and back.
From SOG, turn right and go down World Drive. You can either take World Drive all the way to 192 and exit westbound (Publix will be about three or four miles ahead on the right) or you can cut off some of the 192 congestion by going the back way. Take World Drive to the Animal Kingdom exit. Follow the signs for AKL until you get to the traffic light just past the AK entrance. This will be Sherberth Road. Take Sherberth Road to just past the Holiday Inn and you will see Black Lake Road on the right. Turn on to Black Lake. Black Lake Road comes out on 192 past most of the congestion. The Publix will be about a half mile ahead on the right.
The closest Walmart is on Hwy 27, just north of where it meets with Hwy 192. This would be about 5 miles farther than the Publix.