You're not wrong, better leaf springs AND shocks has made my trailer tow like a dream, specifically off road but on road too.
The main issue with most stock trailer suspensions is it's made for smooth highway roads and for the max capacity weight. If you've ever rode in an empty dump truck or even a heavy duty F350 that's not loaded down, you understand how crappy the ride is with suspension designed for carrying x amount of weight...when it isn't carrying that heavy load.
On my trailer, I didn't want to watch it bounce around behind me off road, much like you mentioned seeing. So, I replaced the leaf springs with longer and softer Jeep CJ-7 springs and shocks and boy is it *the nicest* handling trailer I've ever towed. Around town, going through dips in the side street, over speed bumps and even just pulling into my alleyway where it's quite bumpy...the trailer just absorbs the bumps and is so anti-dramatic back there.
I HIGHLY recommend it, even if you just do shocks, I bet you'd see a big difference, but I do personally believe that changing the leaf springs from the short, high capacity stock ones (as seen in the first pic below, still installed with the Jeep springs installed showing size comparison) is the biggest changing factor to a nice ride.

