Those are significantly nicer components than I was imagining.
The linear rails appear to be properly side-clamped against ground registration surfaces with those screws & dowels, and I believe the rails are name-brand Hiwins. I'd be surprised if there's any serious misalignment-induced binding.
You may actually have AC bearings in the combination ballscrew/motor mount. And your coupler is a zero-backlash type.
I'd remove the motor and proceed with checking screw fixed bearing slop. If there is any you can try snugging up the nut. The machine looks clean but used, so it's possible the bearings are a little worn out. On heavy or fast machines ballscrew fixed bearings take a beating.
After that, the nut, nut adapter, and motor mount screws are now (in my mind) the most likely culprits if there's a mechanical problem. Likely the nut & balls, TBH.
Aliexpress has a few vendors of Taiwanese "C3" double-nut ball-screws around $400-$800 depending on length & dia. I don't know what the machine is, but it looks nice enough under the hood to be worth a set of ground ballscrews if you can scrape up the money.