Without looking at the car or driving the car I may guess that the motor mounts and possible trans mount may be collapsed and in bad shape.

Does the car feel rough in your seat upon start up?
That could mean low motor mounts.
You may want to check the flex disks on front and rear of driveshaft and also check the driveshaft out well.

Check all your suspension components out well and also look at the steering shock/dampener.
You can unbolt one end of the dampener and see if it moves freely in and out with little resistance then it's no good.
A good steering shock will move very slowly in and out while you push it in and pull it out like a shock absorber.

I'd say get the car on a lift and have it gone over and test driven to try to pinpoint the problem.