It's very easy to remove the steering damper and check it by stroking. It should offer near uniform resistance throughout its range of movement. For sure it there is any evidence of leaking, it should be replaced. They cost about forty bucks at the dealer.
Unbalanced tires generally cause a mild to moderate "shaking" at the steering wheel. A "shimmy" is a sometimes violent wiggling back and forth of the entire front end, and a bad steering damper combined with poor wheel balance or a tire out of round situation can cause a rather violent shimmy on 201s. Happened to me a couple of years ago - bad damper and out of round tires (had the tires analysed on a Hunter 9700) - replaced both the damper and got an adjustment from Dunlop on a new set of tires.