The fan switch I think I am right in saying is on the stat housing on these and has a green housing. I think you are unplugging the wrong one as once you disconnect the switch it will stop fans running. It sounds as you have the right switch but you are disconnecting something else as it looks different and has no effect.

So the fans are on even from cold?

As for economy I would give it service first and if still no better get someone to have a look for a tune.

Martin