Ok, here's some things I would do considering that I am not driving this car and limited info that can be had without the actual ride and drive experience and visual checks that can be done in person.
Starting with the small dollar fixs and see what happens will lead to more expensive repairs,
Have the wheels/tires balanced, some places can balance them on the car and may reveal where the problem is.
Drive shaft balanced, if you don't know the age of the u joints it's a good idea to replace them anyway.
Motor and trans mounts, they may look ok, take a jack and see how much movement they have.
You also could dropping a cylinder, harmonic balancer old and has slipped, the list goes on.
Now that being said, take that damper off the rear and see what happens or have you already done that