Usually a new liner will resolve this problem.
Condensation inside chimney liner.
The wetness on the outside faces of the chimney occurs when the condensate migrates through the mortar.
On occasion we have identified the wetness around a chimney to be due to the condensation of the airborne moisture in the exhaust gases discharged by the furnace on the cold chimney walls.
Condensation can occur when ascending combustion gases meet cooler temperatures in higher parts of the flue finally dropping below the dew point.
This is often a problem when the liner used is not the ideal size or material.
You position the branch of the tee so it is facing down toward the floor.
In some cases the heat from the fire will cause the condensate to evaporate.
A chimney cap had been added to the chimney above the furnace flue when the roofing had been done to complement the existing cap above the fireplace flue.
Leaks in the chimney structure a condensation effect also can happen within the chimney itself.
Instead of allowing moisture to evaporate the condensation may just soak the brick interior and then drip down.
As you know if you plug the top of the chimney above the fireplace the flue liner condensation will trickle down to the fireplace.
Then reconnect the flue to the other end of the run.
You will need to protect the insert from moisture problems.