You could also check for a broken motor mount. The side that lifts with the torque is usually the side that will break. Not common, but possible.
Have a friend, someone you trust, get in the car, start it, put in drive and press hard on the break and then start pushing on the gas pedal. You watch the motor to se if it starts to lift up on one side. Don’t stand in front of the car while doing this. 
And does it continue to shake when you put it in neutral and let off the gas when you are going down the road?
And parts stores. There are some good mechanics who work at parts stores, and they talk with a lot of mechanics who buy parts and they are always talking about the weird problems they are having with cars. Your story may be familiar to someone at a parts store. And the advice is free.
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Blessings to all
Jeremiah 29:11 For I know the thoughts that I think toward you, says the LORD,
thoughts of peace and not of evil, to give you a future and a hope.
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