To really test for a broken engine mount you should have the car in drive and the parking brake fully on as well as one foot firmly on the brake. You have to have some resitance load on the motor to check for excessive movement.
Even though you do not have a CEL, you should still check for trouble codes. Advance Auto or Auto Zone will do this for free. You can still pull a code with a scanner even without the CEL indicating an issue as it does not come on until issue is a possible no start or stop running issue.
There may be a bad shift solenoid. A good idea would be to service the transmission by replacing fluid and filter. See if this changes performance as old tranny fluid can really cause some problems as clutch pack material starts to build up. Especially with the mileage on the car, if it has not been serviced n the last 30,000 miles, it certainly needs it. Clutch pack material and contamaitnation can clog shift solenoids and create the symptoms you have. New fluid can clean up solenoids and lubricate the system better allowing it to work withotu resistance. It will also raise the pump pressure as the viscocity decreases over time and that will cause a lot of problems.
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Tuesday, November 15th, 2011 AT 2:05 AM