First off, it is essential that your tongue weight be 10-15% of the weight of the trailer. 13% minimum in my opinion to be ideally stable. Some people notice sway (back and forth movement over a certain speed) when they get down around 11%, so to me 10% for a travel trailer is a little low. Maybe fine for a boat. You really need at a minumum a good guess of your total weight and an actual scaled weight of your tongue or, just to to the truck scales and get all the weights. That will be helpful as well.
You want the weight by axle sets. Front truck axle, rear truck axle, trailer axles together. There are sections to the scales if you are at a CAT scales, or even some other scales. You want all axles weights with each pass through the scales.
Weigh the entire rig 3 times. Once with just the truck, loaded for camping. 2nd time with the trailer hooked up but no weight distribution, the 3rd time with the trailer hooked up but with your weight distribution also functioning, just like you would normally travel. With these measurements, you can calculate your axle loads, tongue weight, trailer weight, vehicle weight, and proper set-up of your weight distribution system.
Second, you need to make sure the trailer is level or slightly nose down. It is more stable like this.
Third, you need to evaluate your hitch set-up and if you have the right hitch. What kind of sway control do you have as part of your hitch? A good integrated sway control system will help significantly reduce the amount of push/pull you get from passing cars and trucks. This is a separate issue from the sway that happens by itself when you go faster.
Sure a 3/4 ton truck will handle it better, but you need to get all this fixed first regardless. Make sure it is set-up correctly with the correct amount of tongue weight, and you'll be much better off. The sway should be gone and the push/pull should be minimized. If you want to minimize it further, then more expensive hitches or a more HD truck will be needed. You just need to fix the basics first most likely.