There's obviously a live feed present, otherwise the light and fan wouldn't come on. I'm rather concerned that the fan is on the same circuit. The fan should be on its own circuit/lighting circuit/fused spur.
A simple test is needed to see if there is a live feed present within the shower casing, but it does sound as though the shower unit has become faulty.
You're quite within your rights to check the shower unit out yourself if you are confident, but, with the possibly unusual way the circuit has been arranged, then I really would advise that you have a pro check it for you.