Flick the A/C on, it'll clear pretty much instantly. Only in town/stop start traffic do you use more fuel with it on, at any half-cruising speed the extra usage is negligible.

Bare in mind, that when A/C is in use it creates a lot of condensation (hence the drip under the car). The internals of the system also get damp, so when you switch the a/c off, that dampness is blown into the car.
To dry the system properly if it's bothering you, have the blowers on full heat for 5 mins with a/c on, then switch a/c off and it should cure the phantom misting up.
However, if you bring some dampness in the car, or even if it rains whilst driving it will always start to get damp inside. Cars are sealed incredibly well now, so the damp air through the vents when it's raining stays in the car.