A vibration or shudder under heavy acceleration or under a load can be
caused by bad or out-of-phase U-joints.  Obviously, the only way for a
U-joint to be out-of-phase is if someone disconnected the drive shaft and
reinstalled it out-of-phase.  I doubt if the center support bearing is
causing the vibration because the vibration would tend to be worse at higher
driveshaft RPM than when starting worse, and you would probably get noise
from the bearing.