Howdy marshcat,

Visually inspect the entire driveline; from the motor mounts to the rear suspension and all points in between. You are looking for collapsed or broken rubber.

Grab a wheel and push/pull like hell. Use a 2x4 as a lever and try to move first one wheel and then the other against the suspension. Does the rearend move around? Is there a broken shock or shock mount? Blown out bushings?

Push/pull on the driveshaft in every direction. Is the carrier bearing siezed? Is the rubber donut OK?

Drop the back half of the driveshaft to check the u-joints. You're looking for worn or frozen joints. Grab the pinion yoke and yank it around.

Post back and let us know what you find.

My guess: seized or sloppy u-joints and/or a broken rubber donut on a seized carrier bearing.

*edit*
I was just thinking about something. If the clutch is oil contaminated the entire car will shudder like crazy when moving away from a dead stop. Usually the shuddering only happens from a dead stop and is worse in reverse than in first gear. Is this kind of vibration you have?
*/edit*

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Mr. Shannon DeWolfe -- I've taken to using mister because my name misleads folks on the WWW. I am a 53 year old fat man. ;-)