HD trucks use a different suspension architecture that don't have the same lateral issues with lift.
At the front, they usually use a solid front axle. With leaf springs and a lower radius arm type link to locate things laterally.
Same with rear, leaf springs and trailing arms to locate things laterally .
If you could describe the side to side motions a bit more and under what conditions, it could help with what to focus on.
You have guys here that say their lifted trucks handle well. It's all relative and people have different expectations. I can say even at stock ride height, with large amplitude suspension strokes, I can feel the rear in particular move around on account of the panhard architecture. Which is why even though I could do an easy minor lift with AHC, I prefer not to in order to keep the suspension geometry optimal for handling. I use to road race so I am more keyed into suspension motions. Perhaps you also are more in tune with what you're feeling.
@Chris Carrera is on the right track. A large lift in the rear done properly will have a relocated panhard point.