Took me a while to diagnose a very similar problem. The wheel bearing on the passenger side was loose. With the wheel off the ground I could feel some play in the bearing when jiggling the wheel. Tightened the bearing until I could not feel any more play. On a test drive the shimmy was almost non existent. Tightened the bearing just a little more, not overtight!, even though I couldn't feel any play from the previous tightening and now no vibrations.
Can a wheel bearing be damaged from being overly loose?
my 2 cents