That photo says it all. Even with the fan off, warm damp air will float up into the pipe due to the chimney stack effect and then cool and condense. Then there's nowhere for it to go.
Eave's would help, changing the vast majority of the pipe to rigid smooth pipe would help, then the water wouldn't pool and could evaporate again, and give a slight fall to outside and make sure any water running down can safely exit the terminal, so use sealant to make the connections.
If you need to find a tradesperson to get your job done, please try our local search below,
or if you are doing it yourself you can find suppliers local to you.
Select the supplier or trade you require, enter your location to begin your search.
Please select a service and enter a location to continue...
Are you a trade or supplier? You can create your listing free at DIYnot Local