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I'm not mechanical in any way but it sounds very similar to what happened to me. It turned out to be some issue with the transmission. There is something called overdrive or something like that where when you go in 4th gear it engages and you get better highway fuel economy. The problem was it would engage but would get stuck and not disengage which would stall the engine (and shake the car violently) when coming to a stop. I took it to a mechanic and they gave me an option. Fix it for 2000 or disconnect it for 100. I disconnected it and I lost maybe 1-2mpg while driving freeway. Some cars have a button to turn off overdrive so you could try that if yours does.
I just looked up your owners manual and you have a switch on your shifter knob to turn off overdrive.