Hrm, we had this problem and the fix was simply to install an exhaust fan ... worked like magic, the room is very dry now wheras before we'd get so much condensation it would literally drip off the ceiling and run down the walls (especially after DD#1's long hot showers). But you've already got one ...

... so it sounds like your fan is doing nothing. Does it actually vent outside or just into the ceiling? If it goes into the ceiling, does it go somewhere there is space or is it blocked by insulation or some other rubbish? Is it an ancient clunker or a fairly new looking one? Approach your landlord/property manager to get it replaced with one that works, use the angle that their precious investment is going to get mould and water damage if they let it go for too long. You could even try mentioning that you have respiratory issues or something and the mould is aggravating it. And put it in writing and take some photos. The longer you leave it the less likely it is to get fixed, but then some landlords are hopeless ...