Yes, tires can make a difference. Tires take a few hundred miles to break in, and if they are brand new, that is likely what's causing your issues. Off-road tires with bigger tread blocks tend to "squirm" more than tires with an all-season or highway tread pattern.
Not every tire works well on every truck, not sure if it's how the suspension is tuned, tread pattern, etc. Your issue may or may not go away with the tires you selected. Post some details on what you bought.
Best thing you can do is put as many miles as you can on the new tires and get them broken in as best you can.