"I'm stumped. For some reason, the FV is cutting out even when it has power going to it."
•No real help on your problem yet, but be aware that the ECU controls the ground side of the FV. So power on the + side, while essential, doesn't mean anything in terms of control.

Maybe the ECU is reacting to an excessively rich O2 signal by going excessively LEAN with the FV???

"Upon starting after about 15 seconds, the FV stops buzzing, but I still have 12V at the connector."
• What is the engine doing at this time? I can only guess it's still running. Because with 12V still at the FV, the Fuel relay (and Lambda relay) must still be energized, which mean the Fuel relay is still getting coil pulses from a running engine.
Running badly? Rich with black smoke? Need more clues here.

How are you measuring O2 Voltage? Disconnected from ECU (open loop)? If not, that might give a clue about mixture, assuming it's not all sooted up already.
See what it reads disconnected, if the ECU is happier that way.
Have you tried another O2 sensor? Or tried cleaning the one you have with a good application of propane flame?

I think we need more background and details. Without knowing what led you to the FV symptom, there are just too many things (for me) to guess at—too much room for off the wall speculation. Like I've done here.
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Bruce Young
'93 940-NA (current), 240s (one V8), 140s, 122s, since '63.