By always on I assume you mean the clutch is constantly engaged?
If the relay is stuck, the clutch should disengage if you pull the relay out. My Vette used a standard micro relay for the A/C clutch, the Silverado is probably a similar setup. Swap the relay and see if that cures your problem. If not, you could have other issues, such as a bad pressure switch, freon charge issue, a bad A/C clutch, or a problem with the control unit.