We have a winner!
Spot on, Alan! I don't know a heck of a lot about GE transmissions, but in the documentation for this one, there should have been 54 ounces of oil in it! I didn't see any evidence of an oil leak - I suspect that the oil leaked out a loooonnnng time before I found this washer. GE's do have a nasty habit of losing their oil when a seal or two fail.
Now I'm patiently waiting for some warmer weather to clean up that gunk and do some further disassembly. Our local hardware store in Ogden is going to assume we're living off the grid when they see how much kerosene gets purchased... LOL