I remember having a similar issue with a SF354, which turned out to be a component had broken (a reed switch iirc). Might be a similar problem to yours, as there are two of them, one for the write protect and another which does who-knows-what. Certainly worth opening up the drive as it is simpler mechanically than later models. This was the rubber belt driven drive, however, if you have a more conventional single sided floppy I am out of ideas.
I tracked down the thread here... (the search on here is a right pain, I can never find anything)viewtopic.php?f=15&t=22308&p=198001&hilit=sf354#p198001
Firebee, Falcon CT60, Milan 040, Falcon MkI, TT, Mega STe, Mega ST + Lots of STs of various flavours