The Hellwig will do nothing to control what you are feeling. Those are for controlling the movements of the tow vehicle lean.
You are correct that that wiggle you are feeling is normal. As long as it doesn't develop into a full blown sway even you are ok. If you want to try to get rid of it you might try the following.
Larry is correct that a low tongue weight will contribute to sway conditions but it doesn't sound as if that is your case, in my opinion.
The type of sway bar you need is one or two of
these sway bars. One is usually recommended for trailers up to about 24 feet long and two if trailer is longer.
These are the cheapest and simplest of the sway control methods.
Barney