The exhaust valve for the system is built into the compressor, however I completely agree with the above, a new one won't fix the issue. The compressor can obviously do its job with air supply, and once commanded it is able to lower the truck. You are looking at the switch, or wiring between it and the ECU.
The switch itself is a simple ground signal, one wire for up, one for down. However, it is built into the Terrain Response ECU, so replacement might be pricey.
Pop the top of the console off, and check for signs of corrosion, or anything getting spilled. Look closely at the Terrain Response ECU, the big green connector that goes into it, and also the big green connector that is below, and clipped to the RH side of the console.
To test, you would need to de-pin pins 4&5 (black/pink and black/white) from C0700(green one into the Terrain Response) and supply ground to pin 5 (BW wire) with the key on. If it then lowers, you are looking at the switch, and it might be worth trying to carefully take apart the Terrain Response ECU and clean the contacts.
You will need to de-pin the wires from the connector, and reconnect it in order for the system to not fault from the ECU being unplugged.