Hi,
You're right, I think the only place to buy sandwiches by the beach is that place you mention, and they weren't up to much. Incidentally, just across from there on the other corner you'll find a small booth which sells drinks; it's run by the army I think (or the Police?) and cans of Coke etc are really cheap to buy. My partner (who's Turkish) always bought our drinks from there before we went onto the beach. It isn't easy to spot as it's tucked inside a little courtyard.........but you'll find it.
The only other thing to do if you want a picnic on the beach is to pop to a local shop beforehand and make some yourself. We lived out there so it was easy to do, but even if you're staying in a hotel you could still make a picnic I think. I'm sure they'd lend you a knife and fork to prepare it, and you can buy all the tin foil etc from the local shop. When we took a picnic I used to buy a loaf or two of fresh bread, and butter it inside lengthways, then I'd fill our picnic basket with boiled eggs, whole beef tomatoes, cheeses, cucumber, dolmades (stuffed vine leaves) squeezed with fresh lemon, a couple of cold roasted chicken legs (you can buy hot ready-made ones in the large supermarkets if you like), and a few peaches to eat afterwards. I'd also take an ice-cold bottle of white sweet wine to drink with the peaches (if you wrap it in wet newspapaer it stays cold), and we'd buy our beers and ice creams from the beach-boy if we were too lazy to walk back to the army shop!
Strawberry