Replacing fan clutch may not change behavior.
Replaced my fan clutch about 5k miles ago, cause it was staying stuck mostly on and periodically throwing the error code for it. This fan clutch design by GM is a weak point of these trucks. It is normal to find it on sometimes as you describe, and my new one does still engage sometimes. No more error codes for me, and engages much less frequently. Annoying sound and a power thief! I did not have the vibration associated with it as you describe, and hence why I thought you were describing a problem linked with bad motor mounts. Should have realized the drone sound you stated, which I often say sounds like a jet engine roar.