Chimney damp patches?

Our house is circa 1860. There was obviously a long standing chimney dampness problem when we move in as both stacks were crumbling towers of dust. However, hubby has rebuilt stacks to exacting standards - flashing, brick work and flaunching. I would have expected the damp patches to dry out (albeit very slowly). But, the damp meter we have got indicates that the damp patches worsen with heavy rain. Stacks are open, so was considering a cowel - but internet research suggests that this won't solve the problem. House was also re-roofed and the stacks must be water-tight. Chimneys are not currently used and I suppose ventilation could be improved slightly but still the rain-related ingress is worrying. Any suggestions would be gratefully received - Thanks