[Osem]

Can't answer the above but if it is condensation, then the warmer weather ought to see it improve. If it's damp from outside, depending on the cause, it will either reappear when it rains a lot or remain pretty much the same. Heaters don't remove humidity, only good ventilation or a dehumidifier will do that. Bedroom cupboards are cool places with little airflow and after a night during which 2 adults have been breathing, there's bound to be a build up of warm moist air which will condense on cool surfaces, especially if all the windows are closed. The problem's also going to be worse if damp washing is being dried in the room for example, or if it's adjacent to a bathroom which isn't properly ventilated at bath/shower time. If you have lots of condensation on your windows every morning that's a good indicator of the problem.

If it were me I'd try to work out what's likely to be causing it and then approach the landlord with the possible solution. A dehumidifier won't cost much and will be easy to install so he may agree to provide one as a cheap and quick way to alleviate the problem and work out what the best long term solution is.

Just one other thing, condensation tends to cause black mildew on surfaces like walls, wallpaper etc. rather than serious fungus, rotting wood or blowing plaster. A few pictures of what the problem actually looks like would help with establishing the likely cause.