Thanks for the input, Arnold - you're not the first to suggest that I might have a screw loose

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The square end of the string shaft isn't slipping inside the tuner gear either, but your suggestion got me to taking a closer look at things and I think the problem is related to the square section of the string shaft being bent in relation to the centerline of the round portion of the shaft.
When I back the tuner key off to lower the pitch of the string, the key moves about 1/4 turn without rotating the tuner gear, and I can move the key shaft back and forth by 1/32" to 1/16" between the two brackets that hold it in place. This creates a space between the worm gear on the key shaft and the tooth on the string shaft gear gear that it should be engaging. As the space gets bigger, it seems like the tension on the string causes the string shaft to suddenly rotate until the gears engage again. That's what causes the awful snapping noise I'm hearing and the pitch of the string goes way flat.
If I pull the the tuner gear off of the string shaft and rotate the tuner key, the key shaft rotates nice and true, and it doesn't move back and forth in between the mounting brackets, which tells me that it's probably not bent, and that the problem most likely lies with the string shaft.
Kinda tough to explain...One way or another it looks to me like I need a new string shaft. And you're right, it'd probably be alot easier to find a decent machinist to make a new part rather than locating a replacement set for a 50 yr old bass. Now if I can give up playing it long enough to take things apart and shop for a new part...
BTW, my Cleveland seems to be enjoying the (hot and) sunny south as much as my wife and I are since we've all escaped upstate NY. It gets to spend more time soaking up the AC than I do...
Thanks again for your help,
Don