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Wheel bearing or what?

Okay, so, new here, but not necessarily new to backyard mechanics. I own an 05 Jeep liberty 3.7l . It currently has 166XXX miles on it. Recently got new tires and they were mounted and balanced, but upon doing so they told me i needed a front right inner and outer tie rod. I replaced both but have not gone for alignment yet - - However, While accelerating to 30mph, if i accelerate fast, there is a “shake” or vibration i feel thru the steering wheel, seat and shifter i think (automatic). If i coast, the vibrating seems to stop. IF im going 40-50 and maintaining speed, i feel the same vibration but as soon as i speed up the vibration goes away.
On top of all those nuggets, today I smelled a burning smell from the rear wheel wells, did a touch test on all of the center of the rims and the rear wheels (specifically the rear driver) were hotter than the front.
So, i got home, jacked up the rear end and did the 3-9 and 6-12 shimmey test, and nothing. BUT i found out i can push the rim into the car and pull it out a small amount. The rear passenger tire does the same but alot less. Still, not alot of play, but enough to cause worry. Lug nuts are tight. And it seems to be moving the rim and rotor, as i see the brake pads being pushed with the rim (NOT the caliper).

Leads me to believe that its a bad wheel bearing in the back, but im not sure - I know i can replace it and would love to save money if I can narrow it down to it being the wheel bearing. I checked the ujoints and they seem fine, no broken parts or play.

Calipers seem to not be frozen or seized. And there happen to be no e brakes on this vehicle currently. As in the hardware was removed and cable tucked away.

Other than that, no weird sounds or grinding or whirring. As well as no clicks, pops or bangs. No weird turning sounds at all. The Jeep does pull to the right a little bit but im sure thats because of the alignment i need. But i dont think the vibrations are because of the alignment. (also, took the car back to the tire shop to have them double check the balancing of all the tires and they said that the tires are balanced properly.

Anyways, if you made it this far in the post, a response would be more than appreciated and I do hope my description leaves nothing to be confusing. Thanks in advance!

Pushing in and out is the standard test for the wheel bearing, but that is usually done with a dial indicator on a magnetic mount because the tolerance is so small, you visually see it. The fact that it is great enough for you to see it is a real concern.

I would first pull the rear brakes and look at the bolts that attach the knuckle/ backing plate or hub assembly to the axle housing to see if they are good and tight. Then you push and pull on the axle to see if it goes in and out. It should take a dial indicator to determine if it is in tolerance, you should not be able to visually see it.