One reason I like to make sure things are set up correct.....I loaded a bobcat in a trailer per a sales reps advice. It was 3-4' farther back than I would have done. Off I went with an 8500 lb trailer behind my 12k lb class 6 dump flatbed. Trailer was all over the road going side to side.
Pulled into an empty DOT scale along the freeway.
Weighed things. Truck was at typical 5900 front, 5700 rear empty weight. Trailer was at my estimated total 8500 on axles. Ooops, I had no hitch weight! No wonder the trailer was going side to side!
Moved the bobcat back the 3-4' where I thought it should be. Trailer axle was down 1000 lbs, rear on truck up 1020, front down 20 lbs.
I was two finger steering after that! Trailer tracked straight!
A BIG truck is not always going to stop fishtail sway!
If as noted, one is talking side wind movement. If the trailer is pulling straight, there is some side to side rocking, I call that somewhat normal. On my travel trailer, no bars there was more rocking, plain WD less, dual cam less yet. I could still pull the trailer at 60-65 without issues in 30+ crosswinds. With an 80+ mph gusts on I205 going over the Columbia river, that gust move me and everyone else in the road a half lane to driver's side. I had no fishtail sway. Everything moved as a unit. That's not to say a few brown stains weren't on my shorts!
I'm a set the trailer up so it can be pulled at 60 no bars etc. Then add the bars/Hensley as a backup. Because if you trailer wants to uncontroll sway move etc without bars, if the bars brake, fall off, as I had happen once, you will be in trouble! I didn't notice the bar fell off for 80 miles up and over a pass, mid winter with a light amount of snow and ice on the road
With this also said, I prefer not to pull trailers over 2x the rgawr. For "MOST" 15 series trucks, the rating is typically a max 4000-4500. So 9000 lbs of trailer. A 10k trailer, I would want an 8 lug 6000+ lb rated rear axle.
Today many 15 series trucks have more power, better gearing, performance etc than the BB 25/35 series trucks I owned and drive in the 70's and 80's. My C6 GM 1500 out pulls these trucks. I still would not pull a 10k box with it! Suspension is too light IMHO.
IF the trailer still wants to sway, a bigger truck will not fix the issue, as I pointed out at the beginning of this post.
A slightly under powered correct chassis truck, is better than ample or overpowered too small of a chassis truck.
Marty