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I am having the same issues right now!!! I was told to do the neutral thing to. My mechanic told me it was my motor mount if the vibration went away while coasting in neutral.
I'm hoping its the tranny mount, but not likely. Most def don't want it to be the driveshaft. The differential has 4 mounting points, which I'm sure are worn after 220k+. Its not wheels and tires, since I've had 4 sets. I'll try the tranny mount first and post my findings. X my fingers...
I read somewhere you can put a wedge in the suspect mount to give it some pressure, drive it with the wedge in place and see if there is a difference. It's too cold out here for me to try, but its a good idea. Also, you can sometimes tell which are bad because the will clunk under normal use. I'm think mine is the tranny since I get the clunk during R and D shift change. I'm sure the motor mounts are shot too.
Probably transmission mount, but could be work motor mount(s) as well.
Thats my guess, engine, trans, or diff mounts, or even the driveshaft. Have someone sit in the car and go from P to N to D, etc while braking and see if you can see anything shifting or moving under there. Pay attention to the diff, trans, driveshaft, etc
The tranny mounts on these cars is fairly soft rubber as are the motor mounts especially as they get up there in mileage/age. It may or may not be the mounts. Could be anything in the driveshaft, rearend or wheel bearings. We've had a similar vibration since we've owned the car for 3 years, let us know what you find.
Mine does the same ish since I got it, tranny has to be bad though, I shift from p to d and BOOM whole car shakes, and when im braking while going downhill the shifter shakes back and forth