Here's what I can say about the steering box. I had/have a heavily wheeled 2004 TJ. Several years ago I replaced the stock Mercedes box with a PSC. As I understand it, the Mercedes box wasn't the best choice for the demands that Jeeps require. I had steering play as you do and the Jeep seemed to have a slight wander. I don't expect a new steering box to fix your wobble, but may lessen it....or not.
I investigated several replacement options. One company had a good reputation, but then was sold and quality changed. I considered buying another Mercedes, but they flat out weren't to be found anywhere. There weren't even any cores to have repaired. PSC was the only sure, high quality option out there, but almost twice the cost. I spent the money for a PSC and never regretted it. The slop was gone immediately. Turns were tighter.
It was a direct bolt on and used the stock steering pump. I did need to change one hydraulic line (mandatory modification) which can be purchased from PSC. The PSC box uses power steering fluid. The stock steering box uses ATF 4. Using ATF 4 in the PSC box will ruin the box. It wasn't that difficult to clean out the ATF 4 in prep for the new power steering fluid. I'm being redundant on purpose about the fluids to be clear.
The PSC box is longer, but bolts up fine to the frame and stock steering shaft. I did go with a Borgeson lower shaft later. I think there's a plastic pin in the stock steering shaft to drill out to allow the stock shaft to collapse further. I'm not sure if my body lift changes any of that.
Stock Box and longer PSC box....