Welcome to the Portal.
Nice build on your trailer, very innovative use and integration of free parts.
I too think shocks would help. Having just the coil springs puts you at the mercy of Newton’s 3rd Law, "To every action there is an equal and opposite reaction". So as the spring compresses as the trailer goes over a bump it also wants to expand once it's clear. It also want to expand at the same speed it compressed.
The shock will be able to slow the movement down and reduce the bouncing. If you can find a shock with variable valving you could dial it into your needs.
The other thing to take into consideration is the coil spring rating to make sure it's in the correct weight category. If the trailer is too light and the spring too stiff it may be difficult to remove the bounce.