Sounds like worn/damaged front end/alignment issues.

Considering your mileage and age, you may be getting a cumulative stackup in wear, allowing the shimmy.

I don't believe it's a balance problem, that would be there all the time, regardless of road surface or cornering.

I've had this problem on a couple of cars, and in both cases, the front end was pretty shot.

Also, if your new wheels have a different offset than the originals, and the line drawn through the steering knuckles no longer intersects with the center of the tire contact surface, that can exagerate the problem of worn components. E.G; If the center of the tread is now outside the line thru the axes, a bump or force on the tire/wheel will increase the deflection force of the bump. Another way to visualize this is to imagine a rod that you hold at the center of the length. If you attempt to push on the rod at your hand it is hard to deflect the rod, you can only push it back. However, if you move a bit to either side and push, you now can easily deflect one side of the rod, adding a twisting motion as well as the intial force vector pushing straight back.

Jim

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14 E250 BlueTEC black. 45k miles
95 E320 Cabriolet Emerald green 66k miles
94 E320 Cabriolet Emerald green 152k miles
85 300TD 4 spd man, euro bumpers and lights, 15" Pentas dark blue 274k miles