Motor mounts won't shimmy the car side to side.
Front TOP torque mount won't do that either. When the
bushings are bad in the upper and or lower torque mount,
you get a lot of engine rocking. I filled mine with urethane.
Vibrates at idle, but a huge difference in braking and
hole-shot.
You can still rock the engine with the factory mounts, the top
one is very soft. When I lean against the car, it moves a whole
three inches with the factory mounts. Barely moves a half
inch with the urethane filled mounts.
You have two other mounts that could be bad, it's usually the
lower transmission mount. Not the torque-rod, the lower
mount.
9858 = Upper Torque
4661H = Lower Left
9903 = Upper Right
9857 = Lower Torque
Check the Lower Left.
Look at how open the upper torque mount is. IT absorbs a lot
of shock and vibration, but it's sloppy when braking and
decelerating. Filling mine wth urethane, took all that mess
out. You can mistake transmission judder for the engine
rocking back and forth when braking.
An issue I have like this with my Mustang, the rims are bent
or out of round. I have also replaced EVERYTHING up front,
except the rack & pinion unit, and it still shakes. IT has to
be the rims.
Another time, a belt was broken in the tire, Camaro.
REAR uses spherical bushings in knuckle, when they go bad,
you can get all kinds of problems. I did not want another
car with them, like my Camaro had, but I had to settle
anyway. I am not looking forward to knuckle replacements
like I had with my Camaro.
Inner tier rods could cause your issue, and tie-rod ends.
If it's not the rims or tires, then it's a suspension component.