You might consider removing the fan unit temporarily and testing it on a bench. Then you'll know if a new one is required. Make a note of the connections when you take it out so that you can rewire correctly.
Vent fans often have Live and Neutral connections and a mains turn on trigger wire from the light switch (Switched Live) - when the fan "sees" the light go on it will run until the light is switched off and then run on for a while longer.
Peccavi
September 2009