your fly wheel or starter is not balanced it sounds like it is your fly wheel because your transmission mount is gone or your motor mounts are gone. either one will offset when the mounts go.
There is a vin# break in that model year for the flywheel. The 8th digit of the vin number determines which flywheel was used. Basicly they used 2 different flywheels that year. I had that problem and after much research i found the correct one at oreillys auto parts. All other parts stores did not show a vin break in the part number.