These came out as an after market accessory about 40 or more years ago. I installed two on friends cars. If you have a volt-ohm meter, you should be able to ring out everything in that circuit. Starting with the fuse, the switch, and the compressor. The ones I installed had 1/8 or 1/4 inch plastic lines from the pump to the rear axle where they teed off towards eack shock. There was also a switch to deflate the system, if memory serves.
Contact the dealer for the locations of these items. I've done a ton of plumbing during the last 40 years, and do not like plastic lines on anything. Could be just an open line.
9 out of 10 times it's the switch that connects to the rear axle. In particular it's the pivot connection (like a hinge). However, running this system without air WILL cause the air shocks to become defective. I have had a couple compressors burn out also due to the leveling switch on the rear axle. On Lincolns the pressure relief valve is on the compressor. It also controls the light on your dashboard. There is also a switch on the older compressors to turn them off. You can substitute the compressor with Cadillac Air Ride they are almost identical and found at most salvage yards.
The reason that you should turn it off before jacking is so that you do not over pressurize the air chamber when lifted so that the rear of the car is jacked up once you let it off the jacks. Some of these have air shocks while others have air bags inside of the coil springs. Those with the bags are made by Air Inds of Lansing on Terminal Rd. They have the air supplied to them via a 1/8 inch plastic air line and if you must repair that line you cannot use a standard union. It must be for plastic or vinyl line as big trucks use. Use of anything other and your asking for more headaches. Like Ben, I've been there and done that too.