You are feeling drivetrain lash.
It's because of two things:
1. Slack in the AWD drivetrain, from all the moving parts that have to work together to get power to all four wheels, which you will not be able to get rid of with any sane amount of money, and
2. Really really really soft drivetrain mounts that let everything move around.
One of the top three upgrades I recommend is a transmission mount, because the oem mount on the WRX is like a piece of bubblegum poked full of holes.