So, I wiped them all down, and commenced to figuring. Here's how to do it:
First, fill a bowl or bucket with enough water to completely submerge an egg by a couple of inches. Begin placing your eggs in the water gingerly, a few at a time. If you plunk them all in haphazardly, you risk breaking them, either from them knocking into each other or plummeting to the bottom of the container and cracking.
Pay attention to how your eggs are behaving in the water. If they float to the top right away and stay there, they are definitely bad. I found a lot of really bad eggs in my basket. One of them broke open, and then I could really tell how awful it was.
If they settle to the bottom, they are good. Eggs that lay flat are fresh. Eggs that stand on end, like the white one in the top of the photo, are a bit older. They are still fine to eat, but they should be put at the head of the line and eaten first, before they go bad.
Since I had put off putting my eggs away for so long, I only ended up with a few good ones out of the whole bunch. Imagine my dismay at having to throw away dozens of eggs, simply because I was being lazy. That's wasteful, and you all know my opinion on wastefulness.
To learn more about cleaning and caring for your eggs, look no further!