I think your setup is fine, it just needs some fine tuning. Your truck can definitely handle the trailer. A weight distribution hitch is a must. We had the Equal-i-zer hitch and it worked great. However, it is pricey and I don't think it is worth the price. You can get a good WDH system and a friction bar sway controller for less.
If you can find the model number or even brand of brake controller, most likely you can find how to fine tune it. With our travel trailer, we had a Prodigy brake controller that had a manual control - a slider that allowed use to manually apply the trailer brakes and it had dials to fine tune soft braking and hard braking. When dialed in properly, it makes all the difference. Trailer brakes that lock up with the slightest pressure on the brake pedal is definitely nerve-racking.
Take the time and spend the money to get your hitch and brake controller setup properly, that includes fixing the 6 or 7 way plug. You driving experience will be much better for it. Also, have those tires inspected professionally. Most likely it is time for a new set. Don't cheap out on tires.
-Michael