Trailer sway

If you inspect your tires after each stop then that isn't your problem; however, you should immediately get rid of those original tires. Manufacturers put the lowest bidder's (ie: cheapest) tires they can get away with. G rated (as suggested) would be my recommendation. As for the sway, sway control set-ups help after sway starts. If you really want to eliminate sway then upgrade your tow vehicle to one that is built for towing and not daily driver. When I went from a 1/2 ton to a one ton there was immediate comfort in towing and no more white knuckles every time I got passed by a big rig.