2012 Peterbilt 386 - Extremely bouncy

Likely multiple issues:
Things that usually break rear engine mounts are severe shock loads, loose bolts and or a combination of the two. If yours is the upside down V mount Pet thing they just give up. Then can drop down and sit on the frame.
Things that usually cause transmission bolts to come loose are improper installation or problems from a previous failure. Also high speed vibrations can cause that as well as crack the flywheel housing.
The type high speed vibration I am referring is usually caused by things like a bent, out of balance, or out of phase drive shaft. Displaced springs in the pressure plate or a clutch problem. Incorrect drive line angles.
Stomach mouth over bumps sounds like major suspension travel or axles twisting? I'll leave that to everyone else. Lots of good posts already.
Just my opinion, good luck.

 

Even though Peterbilt told you the ride height was perfect it would seem to be that they could be mistaken.

Did you get the exact height measurements for your V.I.N. # and measure it yourself?

A harsh ride can also be contributed to over inflated air springs.

Have you checked or replaced your air suspension leveling valve?

 

The air bags that make cab ride rough are behind the cab, ck the adj on the rod to see if putting more air in them helps the ride you are feeling, if the air bags are down with no air in them ck for a crimped or broken line,

 

Thank he's on to something! Saw that on a Mack now that he mentions it. Bad valve.