When you hang the trailer tongue weight on your hitch, the tow vehile drops in the rear and raises in the front, like a see saw. The weight taken off your front axle, reduces control and stability. Sometimes, so much so, that subconscious hand movement (could be caused simply by breathing), can cause trailer sway. The primary purpose of a weight distributing hitch, is to restore some, or all of that weight to the front axle.
When loaded for camping, your trailer will have over 600 lbs (possibly 700) of tongue weight. Is that enough to affect your steering ? To some degree, I would say "Yes".
An Andersen WDH, would be an excellnt candidate for your set up. It's lite weight, has built in sway control, works well with trailers in your weight range, and it's clean enough to throw it in the back end of your SUV, without getting grease on anything. It doesn't use any type of lubrication.