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Will it start in neutral and stay running in neutral? If you let it coast and get some speed built up a couple of miles per hour then you put it into drive will it continue to drive and run normally until you come to a stop and then it stalls?
If you start it and it is in park let sit there does it idle normally no issues? If you then shifted to neutral quickly without stopping in reverse does it continue to stay idling? Then I need you to do the test that I explained try and get the car rolling either in reverse or and drive then put the shifter in that direction will it start to move as if a manual transmission would when you release the clutch and when you come to a stop it starts to shake and and stalls?
Okay this is what I suspected. There is a problem either in the torque converter itself or the input shaft of the transmission. The transmission is essentially acting like a manual transmission with the clutch released all of the time. There's no slippage or torque multiplication in the torque converter itself. Unfortunately the torque converter will have to be replaced and more than likely the transmission gone through and inspected and cleaned and or rebuilt. I have seen this a number of times.