Clunking and car Jerking when shifting and accelerating

I have been experiencing a loud clunk and jerk when I shift from park to rev or to drive from park. This happens also during acceleration. My trans has been rebuilt due to a broken torque converter but had this issue before the rebuild occured. I've had my car checked and I was told by the shop that I needed new engine mounts and a new trans mount to stop the clunk and jerking. I've replaced the trans mount myself and will be doing the engine mounts this weekend. Although I was told that these parts are needed I suspected another issue so I have recorded a video of the differential (because the jerk and clunk were coming from the rear) while I changed gears and It moves upward as the clunk noise occurs during shifting. I'm trying to figure out what exactly is causing this problem and how I should go about repairing it.