OK, this is getting even more strange. I mentioned that removing the Help menu pad completely fixed the problem in one app. I've added it back in, without specifying F1 as the shortcut, and it still works. If I add the F1 shortcut, the problem comes back.

I thought "Great, I'll fix this in my other apps!" But not so fast--I removed the F1 shortcut from the Help menu in another app where F1 was bringing up help, but not context sensitive help. No luck. So I removed the Help menu pad completely. Still no luck. Well, at least it's working correctly in one app. Now I just need to figure out what's different about that app.