If you have a trailer that has a lot of tongue weight or a tow vehicle that can't support much weight at the rear, then a weight distribution hitch will transfer some of the tongue weight forward onto the front axle of the tow vehicle. It "distributes" the trailer tongue weight more evenly across the tow vehicle's front and rear axles which help keep it more level which helps steering effectiveness and headlight aiming.
A sway control can use a friction device or a cam effect built into the hitch itself. The cheaper option and the only one I'm familiar with is the friction type. A fixed component attaches to the trailer frame (via a ball and socket for easy removal) and then a sliding component attaches to the hitch in the same manner. The sliding component is sandwiched between the two halves of the fixed component which can be tightened to various degrees by a hand lever which adjusts the amount of desired friction. It allows the trailer to turn side to side but not suddenly as in a gust of wind. It is recommended that they be removed when backing up because they don't have enough range of travel required when a trailer gets turned to extreme angles. They can be removed in less than a minute or in my case, left on as long as you're mindful of not getting the trailer turned to one side or the other by too much.
You can literally weigh your trailers tongue weight on a fairly stout bathroom scale. Ideally you should have between 10% and 15% of your total trailer weight resting on the tow vehicle. Too little weight is prone to making a trailer sway. A single axle trailer is more prone to sway than a dual axle trailer. As long as the tow vehicle can support the weight, I tend to load the trailer and adjust things closer to the 15% side of the range. You can increase the tongue weight by loading more gear in the trailer further forward of the axle(s) or by lowering the ball on the hitch by even a small amount.
I don't know what a T139 looks like but I've towed a small 2,000 lb travel trailer without either component thousands of miles with no problems. When we got a 3,500 lb TT, we added the WD hitch and sway control and there were a couple days when I forgot to tighten the sway control and didn't feel any difference. Now with the fully loaded FF approaching 6,000 lbs I feel the need for both
I should also mention that if the tow vehicle has soft rear suspension and sags due to tongue weight, than this can cause the back of the tow vehicle to move around (pushed by the trailer) which feels like trailer sway but actually isn't. You mention that you feel it on twisty roads which to me indicates a tow vehicle deficiency. Most normal trailer sway is cause by side winds and wind gusts from semi-trucks for instance. Also speed is a big factor in trailer stability. But it sounds from your description that speed isn't the issue here.
Hope this is enough to get you to the next level of questions