It's that increase in sail area aft of rear axle. A lever.
Different tires with stronger sidewalls are a normal change for those towing with half tons on "P" tires. The change to "LT" means lower ride quality for a given pressure vs load scenario, but all are happier. (The other change is to go to short sidewall tires with a bigger tread patch.)
It'll search and find any slack or slop in the steering system from the driver out to the wheels. Adjusting alignment for quick return to center may help some. Feeling it come on us the thing. When it's a surprise, there can be a long list of non-steering components as well.
IMO, a focus on the rear axle and springs is the place to start. Fir, if something moves back there it has a whipctack effect on the front. (And all these comments are in thinking about a high Miles vehicle).
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