The strange thing is this... I get my handy voltmeter out, and the
"dead" outlets in the upstairs bathrooms all read 96 volts AC. I plug
in a night light, turn it on, and the voltage drops to zero. I
measured the voltage coming into the GFI outlet downstairs at 120
volts. I read the voltage leaving the GFI outlet downstairs at 120
volts. Upstairs, it's down to 96 volts. If I trip the GFI breaker and
test the outlets upstairs, they all read zero volts. So I'm pretty
darned sure this is all the same circuit.

If you are using a digital meter the 96 volts you are getting is either some
leakage from the GFI or induced voltage. There is no solid connection.
Sounds like you need to replace the GFI socket.