I think I know of the supermarket referred to above.
From ALK, drive out of the parking lot at at the first traffic light, make a right. Follow this road to the first traffic light which is 192. Turn right onto 192 and get over to the left lane. At what I believe is the first traffic light, turn left and then enter the shopping plaza on the right. There is a "Marketplace" supermarket in the plaza. It is part of the Winn-Dixie chain, however, because it is in a tourist location they do not necessarily adhere to Winn-Dixie advertised sales (hence they are only called "Marketplace"). Prices are a bit higher than normal Winn-Dixie prices but not horribly outrageous.
My understanding is that if you continue going down 192 another few traffic lights you will come to a Publix supermarket but I'm not certain of the exact location.