Travel trailers tow just fine. We had 60mph cross winds on a prior trailer...we got off the road as soon as possible but no sway. Most problems can be traced to incorrect setup. The biggest issue is getting enough tongue weight. At 12 to 15%, it won't sway at all. This presumes you aren't skimping on the truck. An overloaded truck can be uncomfortable. A properly set up wdh helps but won't overcome a light tongue weight or overloaded truck.
Fifth wheels tend to be much more forgiving because they start around 20-25% pin weight. The big catch is you need a truck capable of handling the pin weight. This usually pushes you to a 3/4 tongue for smaller 5ers and quickly jumps up to 1 ton dually as you get larger.
A 10k trailer will need about 1200-1500 lb with travel trailer or 2000-2500 lb with 5er. Obviously some info on truck and trailer can dial in a suggestion better.