First let me say you’re a good guy to be concerned with getting your hitch setup correct. I’ve seen a lot of folks camping that aren’t setup right and just run it whether they don’t know any better or just ignore It. It’s hard to tell from your pic but looks like you’re squatting too much meaning your weight distribution is off. Looks like too much on the rear axle but that’s only a guess looking at the pic. There’s been a lot of good advice on here, I’ll share my experience and hope it helps you. I tow a 37’ TT that’s a little over 11k loaded. It’s heavy and has a lot of surface area that catches wind. I have a solid WDH and would never tow without one. Crossing bridges and passing 18 wheelers do affect me to varying degrees even with the WDH. The best advice in towing is keep a safe speed where you feel comfortable and have the most control. It’s easy to get caught up in the flow of traffic or feel pressured to keep up. Don’t let it get to you just drive safe.
Looks like you have a great WDH and your truck has some innate sway control built in. If you’re still experiencing excessive sway, as others have recommended, you may want to evaluate and adjust the WDH setup. I pasted a link from the Fastway company below that explains how it well. You may also want to give them a call. I’ve called in the past and they were very helpful. I also pasted a link to the Curt Betterweigh device and one of the several YouTube videos that shows its application. It plugs into the OBDC port and works with your smart phone ... is absolutely worth the money. It’s easy to setup and relatively cheap ($99). Once you have it calibrated you can use it to get tongue weight, weight distribution, set the brake gain and a lot more. I use it every time when hooking up to the TT to make sure I’m where I should be. It’ll give you an idea until you can get to a CAT scale if you want. I inserted a pic that illustrates the weight distribution function from when I was hooking up. I take screenshots every time I tow as references to see if something has changed since the last hook up.
Another option would be make an appointment with a trailer dealer who can set it up for you but I’m sure they will charge for the service. It may be worth it if you don‘t have the right tools or inclination. Sometimes it’s worth it just to pay the guy for their training and experience. Torque spec settings are important and for a TT, the ball head mount screw on mine requires around 450 lbs so make sure you’re right and tight.
Good luck solving the issue. Post back and let us know the outcome.