No, switching tires will not change the alignment. If the tires were balanced previously and wore evenly, this swap will not cause a rebalancing to need done either.
Since it shakes/vibrates, I would try to figure out of you can determine which wheel it's coming from, swapping the wheels to different positions if that might help locate the suspect wheel.
Are the tires on both explorers the same size? If so, or nearly the same actual circumference, you could swap one old wheel at a time back onto the running explorer to see which swap makes the difference, then have a shop inspect (in case it's bent) and balance that one suspect wheel instead of all 4.
I assume you carefully inspected the tires for uneven wear.