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The symptoms you described will often come with loose lug nuts. I would highly recommend checking the lug nuts to make sure they are tight before driving the car again.
If the lug nuts are tight then the problem sounds like it is going to be driveline related since applying torque causes it to go away and it comes back on deceleration. It's possible that the problem could be in the transmission but it's not all that likely.
The problem would more likely be a worn/loose inner CV joint. Power is transmitted from the transmission to the front hubs by the CV shafts, which are shafts with CV joints at both ends and the joints allow the shafts to pivot for suspension travel and turning the wheels. If a joint gets worn or is failing, usually an inner joint, then it will cause a knock or vibration that is throttle dependent.
Loose motor mounts can also affect powertrain alignment enough to cause CV joint issues.