Good info from Jason above. To answer your question, I’ve in the past experienced only the most minimal of sway with very strong wind gusts or being passed by semis. It was tolerable and I was always comfortable going 70 or so.
We are now just completing a 2-week trip to various national parks, and prior to leaving, I installed hellwig anti-sway bars front and rear as well as sumo springs on the rear. I would say on this trip, I’ve experienced essentially no sway at all. I’ve cruised at 80 a few times given the high speed limits in Wyoming and Montana. No sway and comfortable towing. I forget it’s back there other than on hills or when watching the gas mileage. For extra info, I’ve averaged about 7 mpg for the trip with mixed highway cruising, 65-80. My travel trailer has wide-spaced tandem axles, which may help my stability some. If you haven’t been to the CAT scale, I’d highly suggest it as Jason mentioned above, as your tongue weight may be close to or under 10% of your trailer weight.
At the CAT scale, all my numbers are easily in spec, other than my max rear axle weight, which is usually right at the max.