I don't know exactly how the "convenience" feature works on these cars, but that is the first thing I would suspect. Are you able to close your windows and sunroof by locking the driver's door with the key from the outside and holding the key in the "lock" position? I'm not sure what year they started that. My 89 didn't have it, but my 92 does.
I saw the same thing happen on a Passat once with the same convenience type feature. Wound up replacing the window motor/regulator as the ECU was built into that. Somebody here will surely know what part is troubling you.

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Fourings

84 Audi Coupe GT with Turbo engine, EFI (buh-bye CIS), Jamex seats, Koni's/H&R's, e-code lights. 210k and counting.
1992 Anthracite 400E, 161k, "Grey Ghost".
1986 VW Quantum Syncro, 192k, gone to the crusher in the sky.
1989 300TE 172k. (Was mine, now also Mom's car)
72 280SE 4.5 (Mom's car)