It sounds to me like inexperience and an improperly adjusted Weight Distribution hitch.
P tires will allow rocking while towing in high winds. Its the wind pushing on the side of you and the softer sidewall tires flexing back and forth.
Sliding could be because of a weight distribution hitch being over adjusted, taking to much weight off of the rear tires, causing them to slip and slide from not enough traction. Add that to the P-tires and side winds and it could be a very scary experience.
Don't know if you want LT tires. Don't get me wrong, it will make the truck tow better but will give it a hard, hard ride for daily driving. I've seen others not happy with the ride quality from LT tires and go back to P-tires.
Now you said you had no sway, so I won't second quess you, but with the Armadas short wheelbase, you should at least have a Reese Strait-Line system (WDH and Dual Cam HP sway control system), Equalizer 'brand' hitch for sway control, or even an older style Blue Ox Sway Pro hitch. The add on bars are questionable with a short wheelbase as yours.
Also maintain that Armada. I had a Titan and it just didn't hold up so take the extra steps in making sure differential fluid, transfer case fluid and transmission fluid is good and at the correct level.