I found the number 5051 stamped on the edge of the output flange of my carburetor. Somewhere online I found that that is a Tecumseh manufacturing number, and it corresponds to part number 640069, which just happens to be the model number of the carburetor that the OP found to get his trimmer going again.This is starting to look like progress.
Ebay is teeming with aftermarket carburetors that claim to replace that model and several others: 640262A, 640262, 640076A, 640173, 640174, 640026, and 640026A.
Given that these things are priced in the $10 - $12 range, I think that I should start there and not even try to fix the one I've got.
If anybody has a better idea, please let me know.
Thanks, guys. Tecumseh must have sold a blue million of those engines.