Sounds like broken motor mounts to me. A motor should not move when shifted from one gear to another but may move slightly when under quick acceleration (blippin' the throttle as we say or "gettin on it") other wise it shouldn't move.
Carefully place a jack with a block of wood under the motor, loosen mounts from motor, jack motor up just enough to remove old mounts, replace mounts, lower motor onto new mounts, tighten motor to mounts. Make sure there aren't any hoses, wires, lines, etc. that may tear while lifting the motor.