If the chimney is capped, damp appears and goes when is raining, and also damp dust is dropping down- so this have to be condensation !
)
of course if not, since I am working in building industry I never had a condensation problem on the chimney breast- this is nonsense !
I will still suggest to check your chimney stack- if the mortal joint or brick is rotten will absorb rain water which follow down finding way in to the chimney duct and if there is any faulty joint/brick will use this as a way out- and in this case it might be patch on your main room, not absorbed excess will run down, mix with soot/dust and drip down on the bottom of the chimney.
that's my opinion
Salute
MINDEX