A shimmy at idol sounds like a motor mount to me.

One way to check is to have someone press on the brake and give the motor a little gas: once with the transmission in drive, and a second time with the transmission in reverse.

If one (or both) of the motor mounts is bad, you will see the engine sort of jump up, either in the front, or the rear, depending on what gear your transmission is in.

Believe me, you will know it when you see it.

Further, don't rule out either transmission mounts or differential mounts.

__________________
Paul S.

2001 E430, Bourdeaux Red, Oyster interior.
79,200 miles.

1973 280SE 4.5, 170,000 miles. 568 Signal Red, Black MB Tex. "The Red Baron".