Well, you did a really good job of describing the symptoms, but you didn't do so well describing the car. Are we talking about an automatic or standard transmission? are we talking about a four or six cylinder engine? You are asking about a drivetrain problem without giving any information about what drivetrain the vehicle is equipped with.
With all that said,
IF we are talking about an automatic transmission being pushed by a V-6 engine, then there is a
Technical Service Bulletin (TSB) that matches the symptoms you are describing. In this case, the problem is most likely being caused by a deformed or damaged exhaust pipe "A" mounting bracket.
Inspect the rubber bracket for cracks or torn rubber. If the bracket is damaged replace the bracket. Bend the stud on the exhaust pipe if necessary to get the stud to go through the bracket as straight as possible so the stud does not "ground" against the sides of the bracket. (see diagram)
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