Apologies for the long post, but here's my theory on what might be happening.
New tires will often feel a little loose at higher speed until they wear in. This was true for every new set I installed for the first few thousand miles. Your old tires would have had a nice flat wear profile, which gave them a wider contact patch on the pavement, so this could be part of the initial difference in driving feel.
Alright, now for the steering box:
Loose steering boxes of this type should almost NEVER (EVER!!!) be adjusted at the sector shaft. The majority of play you feel is usually going to be in other components that can't be adjusted unless you completely tear down and refurbish the steering box. And even then, most shops aren't equipped to do much more than replace seals. I learned this the hard way, and so are many others because U-tube is full of people showing you how to tighten these boxes, without realizing they're only tightening part of the system and potentially making things worse.
Read Head supposedly does a better rebuild job but they are very expensive compared to a typical NAPA exchange.
I mention the steering box because like others said, if you over-tighten the adjustment stud, the steering will have too much friction in the straight ahead position and the steering will not return to center when exiting a turn or completing a lane change. Combine this with slack that's still there (which could be from any component between the tires and your steering wheel), and it feels like the truck is always trying to swerve on way or another, forcing you to constantly over-correct from side to side.
I can't ride shotgun with you to be sure, but does this sound like the S-swerving you're experiencing?
The sensation can be disorienting because it will feel like some of over-steering is coming from the rear end of the vehicle. This might be why your mechanic suspects rear shocks/suspension. With practice, you can anticipate this effect and learn to drive in this condition but it's very tiring.
You can check for this condition on a straight stretch of road. If the truck is set in a straight line, it will hold that direction with no tendency to pull in either direction. Even on a slight curve or on roads with a moderate camber, it will rigidly hold that direction (because the steering box is stiff, even though the steering wheel feels loose). However, ruts or expansion joints on the road (or even a mis-balanced wheel) may bump hard enough to seemingly shock it partway back to center.