Id say it sounds like a prob with the fridge rather than the food. Ive lived in some places with crap fridges and had similar problems. But to answer your question I store cheese from the deli counter in tinfoil and ham in the plastic bag it comes in. Ham never lasts more than two days in our house anyway but cheese keeps for a long time.
When I lived in Germany I found that the open-air markets generally had the freshest produce. I know what you mean about fruit and veg, even the green-grocers didn't have very fresh stuff, and the supermarkets were dire. I've told dh that I'm never moving anywhere that doesn't have a Waitrose.