How can I prevent a brick exterior chimney from sweating?

Sounds like you are losing heat from your home when the chimney is not in operation. Be sure to seal off the front of your fireplace, if you have one, to keep the warm air from going up the cold chimney when it is not in use.

I agree with the condensation. Room air is escaping up the chimney cooling down and causing the condensation. You need to seal the fireplace or chimney itself to stop the air from going up. Some have used fiberglass insulation against the flue. The even sell air bags that go into the chimney and seal the opening when blown up.

I made a chimney pillow for mine filled a construction grade garbage bag with insulation and shoved it up into opening may need to prop it in there to keep it from falling out. Here's additional info. https://www.chimneyspecialistsinc.com/blog/why-is-my-chimney-sweating/

Sounds like you need another liner replaced

Thank you for your reply and the weblink.