2012 double cab long bed is bouncing in rear

Just purchased. Love the smooth ride up front but when unloaded the truck has excessive bouncing and sways side to side on rough roads in New England. Any suggestions on how to fix this? When I test drove the TRD it was too stiff upfront. If I install new shocks will it fix this? I want to match the feel up front in the back.

 

how many leaf's in your rear spring set ? The rear spring leafs are stiff and the only way to help that would be to replace the rear with a larger leaf pack that has more than like "2" leafes in it

 

I checked and it has 3 leafs the same length along with a shorter leaf. Wount stiffer shocks help. O can go to back of truck and it's easy to push on bed and make it compress