If you can this, open the hood so you can see the engine while setting in the truck.
Start it up and put in second gear,
While holding the brakes, let out the clutch and see if the engine torques overs, engine may die and that's OK.
Then try it in reverse,
Two quick test of the motor mounts, it should torque over some, but not much.
It does appear to be a " under load " vibration. Driveline angles, u-joints, motor mounts could all be suspects. Does it vibrate as bad on level ground ? Is it the speed or the torque being applied?
The left and right turns don't make much sense, but not driving it myself I don't know.
Put the rear up on sturdy jack stands and slowly run it up to vibration speed. Then have a helper do the same while you watch under the truck from a safe distant ( NOT UNDER IT ! ).
