My gearshift visibly moves around 1/4" when accelerating hard or from a stop. (And I can certainly feel it move backward)

Is this normal, or an indication of either a motor or transmission mount being broken?

 

A sure way to test for broken engine mounts (most commonly the drivers side) is to open the hood, put the car in drive, press the brakes, and give it some throttle. The drivers side of the motor should rise some, if it comes up a lot, then you probably have at least one shot mount. It is recommended that you change all three at the same time.
The engine mounts are "real' fun the change

 

Thanks - I've confirmed all 4 need to be replaced and it's going into the shop Monday (Transmission & Engine)