i think the advice given so far is good. I think the clue is in the consistency of the moisture.
When the moisture does bead up, it has almost a sticky consistency
How would rain water get sticky? Something is being dissolved in it. What could that be that would cause it to be sticky? Could it be sap? That's my guess. Where is the sap? Its coming from the creosote in the chimney.
My first thought is you didn't have a chimney cap. You say you do. I'm envisioning water getting in the chimney and soaking through the masonry to the inside. Along the way its picking up some creosote. Try smelling the sticky liquid. My guess is it smells like smoke or like sap or more likely smokey sap.
Where could that water be getting in? the cap, through the masonary or both.
Yeah, lots of speculation.
You should be getting your chimney swept occasionally. Some say once a year. try a different sweep and ask his opinions.
You can change your cap to one with a built in damper, like this
That might help too.
Post some pics and if you find a solution post that too.