Your truck is probably overloaded towing that much weight. You may have either not enough weight on the tongue which will induce a sway, or trailer tire pressure was uneven, which will induce a sway as well, or you had enough tongue weight, but that put too much downward pressure on the hitch ball (particularly on short bed trucks) which lightened the front end of the truck. Your truck tires may also be underinflated or overloaded.
I would bet inadequate tongue weight on an overloaded truck with a hitch system that was also overloaded. You were lucky on your earlier towing experiences.
Your 1/2 ton might tow it, but if it's grossly overloaded, it will wear more quickly and if you have an emergency situation, you will not have the safety factor of a heavier truck and open yourself up to a lot of liability in the event of an accident.
Sounds like budget is an issue and a replacement truck might be out of the question. You can do a few things to your truck to help, such as over load springs in the rear, but the best thing if you wish to use this truck would be an equalizer hitch(probably in the $350.00 range) which helps distribute tongue weight to both truck axels and also back to the trailer axles. This, with an anti-sway device would make for a much safer and more stable rig, However, overloaded is overloaded, and if you exceed the towing capacity of the truck or the GCVWR (Gross Combined Vehicle Weight Rating) (truck and trailer plus load combined), a better hitch will help, but not solve it completely. Keep in mind that the GVWR (Gross Vehicle Weight Rating) for the truck includes passenges and other equipment you carry plus the weight on the hitch. I would bet you also carry a lot of tools, fuel, etc in the pickup when towing the trailer. The GVWR on a standard 1/2 ton is around 6,100# - 6,400#. Which leaves about 1,800# for people, equipment and trailer hitch weight. Put 1,000# on the hitch ball that far behind the rear axle and you may have a hard time stopping quickly on a curve if the road is wet without a jack knife situation. This could be worse yet if the trailer brakes are not properly adjusted or not working completely. Some dual axle trailers only have brakes on a single axle. If it's heavy enough for dual axles, it's heavy enough for brakes on each axle.
If you could borrow a 3/4 or 1 ton truck just to try it with the same trailer combination, I think you would find a significant difference.
Good luck and be safe.
Joe