Lude Generation

My 5th gen started shaking like crazy when accelerating hard, and when going faster than ~50 mph. The whole car shakes, but the steering wheel starts shaking a bit sooner than the whole car.
It only does it when I press the accelerator hard, and gets progressively more intense the faster I go. If I only press the gas pedal about one quarter, it doesn't shake, but any more than that and it starts... It's like driving on a bumpy gravel road, but the shaking is side to side instead of up and down.

When I let go of the gas, the shaking stops instantly
When I brake with the engine in first and second gear, I can feel slight knocking/thumping, like driving a bicycle with a flat tire.
When in neutral, the engine revs up normally up to the rev limit with no problems.
The tires are new and centered, done it yesterday.
The faster I go, the harder and faster it shakes.

Could it be the driveshafts? or perhaps the transmission?

Sounds like a failed rear engine mount.

Sound like a drive shaft to me

I had a broken driveshaft cv joint which caused a shaking when accelerating, Rich at PA replaced it and problem was solved, but agree it could also be engine mount related.

Check the drive shafts, front and rear engine mounts, front suspension arm bushes and inner and outer track rod ends, as a starting point.

My hunch is on driveshaft though.

I had the same problem turn out to be drive shafts

worth checking the condition of all the engine mounts - 5th gens seem very prone to fubared front mount - so id start there
although your issue sounds like driveshaft like many others have said.

Is there a way to check the driveshafts at home?
Raise the car, remove the wheel and try spinning/turning the wheel?

You just need to lie under the car, wheels on the ground, grab the driveshaft and try to move it. Compare one side to the other and you'll be able to tell if one has too much play in it.

what about the trailing arm bushes ?