The late model deck mounting is very secure and rarely gives much trouble, just make sure that the pins on the rear mounts are through the holes and fully home.
Pull the bonnet off and engage & dissengage the PTO a few times watching the engine.
If it looks like it is shaking, check the mounting bolts for tightness & chassis for cracks eminating from the mounting holes.
Have a close look at the engine pulley. check that the center bolt is tight and that the pulley spins true
If all is well then pull the deck off.
Flip it upside down and check the 4 legs of each spindle housing for cracks.
While it is upside down grab the tip of each blade and try to rock it up and down, you should not be able to .
Pop some 2x4" on the ground and flip the deck back right way up.
Slip the belt off and spin each spindle looking & listening for rumbles.
Do the same with the two idler pulleys also check them for flat spots, even the slightest flat can make the deck dance
Also check your belt carefully for splits and more particular for missing bits. these will also bounce the deck a bit.
As for the deck height, you adjust it by moving the lifting rod to a lower hole then balancing left right with the slip joint on the left side of the deck.
If that nut comes loose the left side will drop quite a bit and this will also make the deck shake a bit.
Once you have it level left right the songle draw bar at the front will need to be shortened to keep the right attack around 5mm.
This set up is prone to excessive wear at the hole where the lift rod connects to the lift bar and also at the other end of the lift bar where it pivots on the tractor body.
I am forever welding them up & redrilling the holes