Lots of good info here.
Yes a wdh and anti sway can/will improve the towing experience but you may as suggested, not have enough tongue weight, because even if you're a little tongue heavy, that truck should handle it.
Every brake controller I've seen for almost 30 years is adjustable to an extent. You first have to adjust "plumb" for lack of a better word on the controller as the older controllers are inertia activated and the "bubble" like a Mercury switch needs to be at the right angle.
Are you able to gently apply the brakes manually? If they immediately lock up with just a little travel on the lever, then the controller may be faulty.
Good news is the trailer brakes work well if you can lock them up.
Knowing you're on a budget, I wouldn't go whole hog on an expensive new hitch setup until you figure out the problems. Under normal conditions you should be able to pull it with little sway or bucking.
Lots of rvnet pros on here in WA. Petition the group, maybe someone could take a look at your setup.
I'll give you a hand if I can work it in and you're around south King County.
Having someone relatively qualified and impartial (not looking for a paycheck) will do volumes for getting you set up good.
You got a pretty good truck sounds like and a good trailer, jsit needs some tweaking.
Good luck!