Trailer wagging

I would be very concerned about a trailer that wags like a dog's tail without a WDH. The WDH may mask the real problem. First, I would check to see if the tongue weight is between 10-15% of weight of the trailer. Then I would check the alignment of the axles to the centerline of the frame. A professional could also check the axle alignment for you to make sure the wheels are tracking properly. Some trailers that are short coupled will inherently be unstable, but that is probably not the case if the tongue weight is correct. A short coupled trailer will magnify the imperfections because the distance between the axles and the coupler is so short. If all of the above are correct and the frame is relatively straight, it should tow good, excluding external forces such as cross winds. Once you get it towning properly, then a WDH can be added for extra insurance. The reason I didn't mention the tow vehicle is because the OP mentioned that his tow vehicle towed a much heavier trailer with no issues.