If there are patches of damp, there may be bridging between the outer and inner wall allowing moisture to wick across,

Possible causes of this are debris in the cavity, cavity wall insulation etc. Often happens below windows for some reason.

Painting Thompson waterseal on the outside wall worked for us, along with having the cavity insulation removed.

Leaking gutters also can cause this, or make it worse if the leak runs down the wall.

All these things are fairly pricey fixes and your landlords responsibility.

Edit Just read you cleaned the gutters out, that's a good start, Also the state of exterior pointing can cause issue.