Does it have aircon? If it does a switch change will not fix it. Not sure but it may be designed to stay on, it helps ensure the car windows stay mist free. Never ever tried to turn mine off.
Check the condition of the fan relays inside the engine bay fuse box (Two green ones). You can swap with another relay of the sme colour for troubleshooting purposes.
Check around the relay bases for signs/smells of burning.
Mine had exactly the same problem except mine hada leaf stuck in the blower fan blades and the thing was starting to sound like a rusty hamster wheel and was driving me up the wall.
I was just about to start ripping the blowers apart (again) when I saw the aircon light on and had one of those 'I wonder if...........' moments - bingo.