If you have enough load in the bed to drop the rear 2 inches, even if some part of that is the tongue weight of the trailer, I would not think it excessive and would not expect to find controlling the load a problem with standard suspension, which makes me wonder just what you are experiencing. If the bobbing is up and down versus side to side, I would have to ask if the bobbing is road induced, or if it bobs around even on relatively smooth flat ground.
Side to side shifting is usually tongue weight imbalance, and you would need more tongue weight. The same condition can also contribute to up and down bobbing, but the shift is more circular than linear. Usually if you have something between 250 and 500 pounds of tongue weight, you have plenty, depending on how much the trailer weighs altogether. If you have a 1,500 pound trailer and the tongue is light enough for you to lift bodily (and easily), it is probably too light. If the tongue weight takes your truck straight toward the road, it is probably too heavy for the vehicle as built.
If this is just a case of once you hit a bump it just keeps on bouncing, you need more damper. Change to adjustable shocks or another shock with more damping.
However, since you are only dropping the suspension by 2 inches fully loaded, I think heavier springs or air bags are not the answer. It sounds to me as if you are carrying the load, just not controlling it. Try loading the truck without the trailer attached, then attach the trailer and see how much of that drop is attributable to load and how much to the trailer. If the trailer makes a good part of the difference in compression, I would look at an equalizer hitch, which will bring your trailer load back toward level and help support the weight of the tongue. That alone should take care of the bobbing all other things equal, but if it doesn't, trade up your shocks.