First off, it's hard to tell what you have there from a picture that far away and with bushes obscuring the bottom of the trailer.
I suspect that your problem is simply CG related. Center of gravity is what is at the core of a trailers towing behavior. Moving weight from the front to the rear or vice versa is simply adjusting the CG on one of three axis, which is the fore/aft or longitudinal axis. There are also the lateral and vertical axis to consider.
It appears to me that you have a very tall and narrow rig. If the vertical aspect of the CG is high on the vertical axis while the lateral axis is narrow (a narrow wheel base, i.e. short axle) then any bump or force applied to one side of the lateral axis will push that high riding center of mass (CG) laterally, which then runs into the resistance of the tire and or shock/spring which will then bounce it back again in the opposite direction. If the CG is high enough and the lateral axis short enough then you can develop an oscillation that feeds upon itself.
That's just a guess looking at the rig from the one picture from afar that you presented. A closer look from a side view and rear view would help to clear things up a bit.
P.S. Remember that the CG is a single point in space if you will. It sits in a position measurable on all three axis and is totally dependent on your rigs design and its load. CG not only effects the stability in towing that is demonstrated by the video linked in the post above, but a high CG effects how easily your rig will roll over on a slope or in a turn with G forces applied. In a static situation (no inertia from movement) a trailer will roll over only when the CG goes beyond the point directly over the pivot point (down hill tire).
