At 36 K it is unlikely your shocks are worn. Bouncing the car will tell you nothing. We did it in the eighties and it sort of worked on hydraulic shocks, good for nothing on today's cars with gas filled shocks.
Squeaking sound will be your polyurethane bushes. Are you aware they need to be greased on a regular basis? About once a year will do it, use lithium based grease.
Who installed control arms? Is it convenient for you to go back to them so they can check front end. How long since last wheel alignment?
Nuts and bolts should be checked for tightness as well as excessive movement in bushes.
Also, it is to early for engine mounts to be worn, are you rough with the clutch. You accelerate hard I believe.
Have you checked tyre pressure to eliminate possibility of puncture? What pressure do you pump your tyres anyway? Do you have aftermarket wheels and wider tyres ? What size wheels and tyres?
Don't know if you rotate wheels on a regular basis or at all, if not, your rear tyres will have more profile so swap them with the fronts and se if that changes anything.
Did you have heavy load in the boot when noticed floatiness? Guessing not. Was it wet?
It is quite normal for the car to wander a little when it gets to longitudinal roots.
Without seeing the car it is hard to diagnose the problem. These are some guidelines you can follow to narrow the search.
Heard many complaints about Pedders, but in the end it all comes which one you get to. Not all of them are dodgy.
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