I noticed you said you had 20" tires? This may be part of your problem. Those tires are usually low profile for street and mostly for looks. If this is the case change them to a heavy duty LT tire like the Firestone Transforce.
Dont listen to those that are telling you you need more truck, that is pure baloney. What you need is to get this truck and trailer set up right. If you do so you probably wont need expensive sway control geegaws either.
Those that have told you to get the weight in the right place in the trailer and measure the "squat" of your truck before and after hitching up are on the right track. Also the guy who pointed out that the "bounce you are experiencing is a Weight distribution problem is right.
This problem is solvable WITHOUT resorting to buying a new and bigger truck. Good Luck.