I had a similar problem a couple of years ago in a rented apartment.
When I first moved in, I was advised to keep the air conditioning on low for about an hour a day. When I complained about the damp and the mould, I was told to leave the AC on high and 24 hours a day. Nothing made any difference.
My landlord called in the builders (as it was a relatively new build!) who advised the place would stay damp and mouldy unless I left the windows open all day - not great advice in this day and age! I was also told the fact I had clothes drying in the house was the cause of the problem.
I did everything I was told - AC on full blast, windows open in every room when I was there (even in the middle of winter!), stopped drying clothes inside and the damp problem continued - mould on clothes, shoes, bags - everything.
I eventually had to break my lease as the smell in the house and the smell off my clothes was just disgusting.
Since then, I've lived in older houses etc - doing the same thing i.e. windows closed all day, clothes drying in the house, and apart from a small bit of condensation and a slight bit of mould occassionaly on the windows, it is nothing like what I experienced previously.
I had dehumidfiers and crystals to dry out the place and to this day I am convinced that the cause of the damp was structural but the builder wouldn't even entertain this idea.
In a normal environment, there shouldn't be mould growing on your clothes and unless the landlord is willing to look into the problem, and try and solve it, the only solution may be to move out, like I had to.