They may well need replacing, but they are unlikely to be the cause of shimmy. The steering damper, perhaps, but shimmy is far more likely to be a problem with free play somewhere in the steering system, as stated previously, starting with the swivels, and after that most likely tie rod ends or suspension bushes, especially the panhard rod ones.
And there is likely to be multiple minor issues that all add up on a vehicle of this age.
John

