Why does the bathroom exhaust fan drip during the summer

I have an American Standard Cadet 2-handle faucet which does precisely the same thing. And don't feel lonely, I do the same as you too. lol I bet the final pesky dripping has to do with the associated residual layer of water on the inside of the pipes after the handles are turned off. Just like an upended ketchup bottle which is empty enough to no longer squirt, but left inverted will eventually cease dripping once gravity has removed the layer of ketchup from the inside surfaces. I realize this is opinion, however I've pondered the pesky drips while sitting in my tub many, many times over the years. That's pretty personal experience...

Condensation occurs when warm moist air contacts a cooler surface. This typically occurs inside the duct, but can occur outside the duct in hot humid climates when exhausting relatively cool, conditioned air. Condensation can be minimized by employing insulation to reduce the temperature differential. It can also be managed by controlling where the condensation collects and drains, such as sloping the duct towards the outlet so any internal condensation drips outside.

In the case of external condensation, controlling via collection and drainage is difficult. Insulation is the best defense. Using closed cell insulation minimizes the chance humid air will reach the duct surface, but if any does, it is trapped by the impervious nature. Using open cell or fiber insulation allows humid air through, but also allows it to escape when conditions change.

In constantly humid conditions, closed cell insulation is a better choice, but if there are cycles of drier conditions, open cell or fiber insulation is a better choice.