I have a 2003 Saab for some reason the air conditioner continues to blow even if the car is cut off and it runs dowm my battery.

You have to change the reele in the circut board

I have the same problem, The fan runs until I disconnect battery. when I reconnect battery it is off, until I put the key in, and it starts and stays on when I remove key. Obviously an electrical issue. I replaced the resistor under the cabin air filter, and it changed noting. I also checked it with the resistor out of the circuit, and it acted the same. So it must be something else. Good luck.

This problem is documented. I had exactly the same problem. The fan speed controller is bad (at least if you have the Automatic Temperature Control). It is somewhere up under the dash or on the firewall on the passenger side. It is easy to change. Not a cheap part, though.

By the way, until you get the fan controller part fixed, the easy fix to keep your car and battery working is to pull the fan fuse for a few seconds, then plug it back in. It is much easier than disconnecting the battery to save your battery.