the driveshaft does not support the differential. If the diff is shaking, check the mounts.

the driveshaft may be a root cause though. If the joints are toast they'll try to shake the diff and therefore wear out the diff mounts, but im surprised you didnt have a crazy driveline vibration if that is the case. Either way, take down the front driveshaft and make sure the joints are smooth. If they're sticking/ratcheting to move, you need a new shaft as well.