This is the back of my MIL's fridge:
Thanks to the helpful suggestions from y'all....I pulled it out and did try just now to clean it. We'll see if it works or if I cleaned it correctly. I did the best job I could vacuuming the plentiful dust and dirt away.
One problem with going out and buying a new one is that the space there is a little tight - a couple inches on top and maybe one short in width. We'd hate to buy a new somewhat compact refrigerator only to redo the entire kitchen not too long after - that is one of my wife's current pipe dreams.
We had an absolutely tremendous kitchen in that rat-house that we rented. It had counter-space galore, a bar, two sinks, etc. And we knew all along that we'd be facing the agony, no matter where we'd move, of most likely a less functional kitchen. So we are having enough *adjustment problems* already without the food spoiling every two days. Plus it would be nice to have a water and ice dispenser on the door again. It's the type of luxury that once enjoyed, you simply can't give up.