Status Report
I ran bunch of cold water into the bucket to flush out the bad, and then started taking the eggs out, rinsing them, inspecting them, and putting them in a new batch of water. I found two cracked eggs in the top couple layers, which supports your theory, Phil. I threw those away of course.
As I was rinsing the eggs, I had the thought that I might now be "washing" the eggs, and that's supposed to be bad, so I've split my initial experiment into three new ones:
1) I'm putting a few dozen of the most recent eggs in the new batch of lime water to see if the "washing" has a negative effect. I'll try to remember to pull a couple out every week or two to see how they fare. That will have to wait a while, though because experiment 2 is...
2) The 8 dozen I didn't put back into the water I rinsed lightly again and put them in cartons in the fridge. These are the oldest, so anywhere from 3-7 months old, stored in lime water almost the entire time. (They didn't go in the water the same day they were laid). We'll eat those first, while storing or sharing the fresh eggs that come from the chickens. That'll take a few weeks to go through!
3) The third experiment is that I cracked open two of the oldest eggs, and scrambled and cooked them. One yolk fell apart, put the other one stayed together, and there was no bad smell. They were a little runnier than fresh eggs, too, but they tasted fine. I only ate a couple bites to try to avoid getting sick if there was a problem. [EDIT TO ADD: No ill effects! I posted as such down below, but I wanted to put the update here in case someone misses the later reply.]
I don't think the eggs cracked from stuffing so many into my bucket, though that is a bit of a concern and I'll be more gentle in the future, now that I know I can fit ~10 dozen eggs in the buckets. I'll probably cut that back to ~8 dozen or so.
Thanks for all the input, folks. Like I said, hopefully I'll be back with some updates!
