Start by coming back and providing enough information to get sound advice.
In the meantime, fundamentally, yes you can put heavier rated tires on the rear without any issue.
Considerations, if 4wd they need to be the same diameter or close if you use 4wd at all.
You just now limited your tire rotation options unless you run em backwards with the heavy tires up front when not towing (like in the winter).
Sounds like you don't really know the source of your sway issues, if you have any issue at all. There's a difference between being blown back n forth on the road and trailer sway. It may feel the same to someone inexperienced but, in short, if the "sway" is caused by something in particular like cross winds or passing a semi then you don't really have a problem. But if the trailer starts tail wagging at a certain speed every time, it's typically a tongue weight issue. If the tail wags more than it should with wind/semis but comes back in line on its own then soft tires could difinatley be part of the problem.
If you have everything else set up correctly and are pretty certain it's soft tires, air them up more and take a little test drive. If putting extra pressure in the rear tires helps considerably then you'd benefit by having heavier tires. And no you won't blow out a P rated truck tire if you run it up to 60psi. I don't reccomend doing this for extended periods even though I've done it a lot in the past when overloading a half ton with P tires.
Better yet, provide all the pertinent info on your setup and let the smart people here guide you through set up options. May save you buying a pair of mis matched tires....
