The relay for the fuel filler solenoid is in the BECM, on the power board.
If disconnecting the switch doesn't stop the 'buzzing' you hear in the BECM then chances are there is either a fault in the wiring to the BECM, or on the power board itself (not the end of the world - they are swappable without interfering with programmed vehicle settings).
If the solenoid was constantly activating then it sounds like either a short in the wiring or a problem in the BECM that's caused it to begin with. Is the buzzing still there?
If you disconnect the filler switch and the solenoid, then I don't see any reason why you can't manually access the fuel filler - either by the release cord, or the method you mention.
If disconnecting the solenoid makes no difference to being able to manually release it, then it could be that the motor in the solenoid has burnt out.