The first thing you need to do is raise the car safely and do the four P's, Push, Pull, Pry and Prod. Look for anything loose. Even with worn out shocks and steering damper, if everything is tight, you shouldn't get a violent shimmy like you describe.
You also need properly balanced tires. Many shops will balance with weights ONLY on the inboard lip. This is static balancing and will prevent the wheel from jumping up and down. You need a dynamic balance with weights on both planes to prevent the wheel from "wobbling."