A broken motor mount will allow the entire engine to twist beyond the normal range of travel when it is placed under load (like when starting from a stop) Depending on the direction of the crankshaft rotation, the engine will twist one way in a forward gear and the opposite way in reverse. The motor mounts limit that movement unless they are broken. If the mount is broken or loose it can allow the engine/transmission to twist under load and bind the drive shaft. Look for loose bolts or cracks around the frame near the mounts. If you open the hood and have someone let the clutch out until it just starts to grab with the Jeep in gear and the parking brake applied - you'll see that twisting motion. I don't have to tell you to watch from the side and not in front of the vehicle.
Some rocking is normal, the mounts have rubber cores to allow a little bit of flex. A large amount of movement indicates that something is broken. The transmission mount is basically just a rubber block with a couple of threaded studs. The purpose of the transmission mount is to hold the back of the transmission up and it's not as critical as the motor mounts but all three work in conjunction to support the engine/transmission.
A broken mount wouldn't change with use, it's either broken or not. If the problem gets better with use then I wouldn't suspect a broken mount.
I suggested the motor mounts because of the violent shaking but your statement about it getting better with use tends to point back to the clutch.