Why would the light be on when the switch is off?
A &quot;short&quot; as you call it seems unlikely since that should cause the breaker to trip.
As a next step, you should SHUT OFF THE BREAKER, and then check the switch since you already seem to believe that it may be bad.  Yes, switches do go bad and so your belief is reasonable at this point.  If you have a VOM (Volt-Ohm Meter) checking the switch is simple enough.  If not, you really should get one but another approach is to simply change the switch (new ones are inexpensive) and see if that resolves the problem.
If that turns out to be the problem, then you should be able to replace the switch and move on.
Otherwise, more troubleshooting is in order.