The airflow is controlled by an electronic "Blend Door" which moves as you select different areas that you wish the air to go to.
It is also controlled by vacuum from the intake manifold....there is a small pipe coming off the manifold to the underside of the dash...if this vacuum pipe splits or comes off what tends to happen is all the air comes out through the demister vents.
You say you can hear the blower motor speeding up when moving the blower speed switch so that should rule out the blower resistor pack which sits behind the glove box and has a habit of blowing....usually not working on position 1, then not on position 2 etc.
There is also a metal grid inside the cabin up against the firewall below the glove box that re-circulated air goes through. This grid can get blocked up but this should give you a gradual deteriation of re-circulated airflow.
You should definitely not hear your internal blower motor running when the engine is turned off and you should not get a flat battery while ignition is off.
Are you sure the blower you are hearing is not the internal blower but the electrical cooling fan in front of the air-con radiator?
You can download the Service manuals for your KJ here....2006 will be close enough for your 2007 model....should give you a description on how the blower and air vents work...also has circuit diagrams in case you have to trace a short somewhere: http://www.colorado4wheel.com/manuals/Jeep/KJ