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The transmission exchange to the Park Avenue might seem like the easiest of the two swaps, and considering that the PA probably has higher value in the resale market it would also appear to be the better choice. The transmission from the 2000 GS is likely the heavy-duty model (4T65E-HD) and would normally be a good candidate for exchange.
However, there are a few items which may cause problems. The final drive gear ratio of the GS could be different than what is installed in the 2003 Park Avenue. The GS is probably a 3.29:1 final drive ratio, whereas the PA could be 2.86 or 3.05:1. If this were the only problem it could be overcome by re-flashing the PCM after the swap, but that adds to the cost of the project.
Possibly of greater concern is the difference in valve bodies between the two years. 2003 and up 4T65E transmissions used a different pump assembly, a different scheme for line pressure control (different solenoid and different PCM control modes), and includes a fluid pressure switch for control that was not present in pre-2003 units. Thus, the transmission is not simply "plug-and-play" between the two vehicles. Exchanging the valve body and channel plate between the two would constitute more effort than a straightforward rebuild of the transmission in the PA.
What exactly is wrong with the engine in the GS? What exactly is wrong with the transmission in the PA?