When I shift gears the cars shakes abruptly kind of like dumping the clutch. Could it be the throw out bearing?

I don't quite know how to describe it but it feels like the motor is disconnecting abruptly.

The car has low miles (~40k), and has mostly been garaged it's whole life so I'm wondering how the motor mounts can wear out.

Its intermittent but it happens often when shifting gears. Especially from 1st-3rd.

 

could be diff bushings in the back

 

It is likely the decade-old Rear Differential Carrier bushings.

 

The rear diff would make the front of the car shake?

Are these the same bushings that make the thump in the rear when shifting fast?

 

That is my guess. Yes!

 

Could be the clutch having a rough engagement too. Maybe slipping