The compressor should not run with the ignition off. In fact, the system should bleed air down when you shut the ignition off and leave the car. Most likely you either have a faulty height sensor somewhere (there are 3 of them on the four wheel system, one on the two wheel system), or the compressor's relief valve is not operating properly. The compressor and valve is located directly behind the right rear tire under the car. Once you shut the car off, run around the back of the car and get your ear down by the bottom of the fender. Within a few seconds, you should hear a click and the car should drop down. At least, that's the way the two wheel system on my Imperial works.
There is a self test for this system, but it requires access to a scanner.
These air suspension systems are a real pain in the neck - when they fail, you can't get new airbags any more. Any kind of problem with the system usually means retrofitting to normal shocks, struts, and springs. The rear axle also needs to be swapped out. Big job, but all the parts can be sourced from a Dynasty if you go the used route. IIRC Strutmasters makes a full conversion kit which may not require swapping the rear axle.
Looked over my info again... there are height control solenoids on each front air strut. If these release but the relief valve on the compressor doesn't, the front would stay up. Same thing if the situation is reversed, but it would not be likely for both front solenoids to fail. One side or the other should get stuck in the air.
At any rate, it would be helpful what exactly happens when the system malfunctions.