The ply ratings are ratings, not the number of plies, and, have nothing to do with bouncing per se, that's typically too high a tire pressure for the load rating of the tire vs the actual load.
In the old days...the number of plies WAS the ply rating....but its pretty much just an advertising thing now, as some people still look at ply ratings as if it meant how many plies the tire had, etc....even some old school tire shop guys still use ply ratings, but they are pretty much useless now a days.
So, for example... if the truck sits with 1,200 lb per tire, and the tire's max load is 3,500 lb at 50 psi, and you have 35 psi, you are inflated to support a ~ 9,800 lb truck, etc...and it may be a bit bouncy.
Now, if you don't mean bouncing from side to side when changing lanes...you mean its "wishy washy"...like the sidewalls are not stabilizing the truck, and its wallowing, etc...that's sometmes just how that tire handles...the first all season tire I ever tried, decades ago...I drove away from the tire place, went to change lanes, and freaked out because I thought the tires were all but flat the way the damn thing wallowed...albeit the tires were OVER inflated, and less air only made it WORSE...so I took them back...a few others, same thing...just poor tires for sharp handling.
Now, I never had the open country AT's...but I never heard anything about bad handling for an AT...hell, anything's better than a grabber...so, I'm guessing over inflation?
And or your shocks are shot...and the new tires are exacerbating the previously masked problems, etc.