We bought our meat, fish and vegetables from butcher, fishmonger and greengrocer..... The downside is that the shop takes longer and requires planning.
If we had any of these in Forest Hill I would gladly give it a try. Since we don't I will continue to buy fish from Sainsburys (vacuum packed means I have never had a problem with freshness, probably fresher than a fishmonger where the fish is left out all day).
I don't think supermarkets have the monopoly on keeping Vegetables and fruit on display that will not last long. I found the same problems with plenty of produce when we did have greengrocers in Forest Hill (20 years ago) or when I had a greengrocer round the corner (in Edinburgh). I reckon the freshest fruit and veg comes direct from a farm or is air-freighted, buying English produce in a supermarket is worst because too often the produce is picked when ripe and packed for fast consumption but it takes too long to reach the shelves.