From doing some Google searches It seems like your Sante Fe is only rated to tow 3500, and 350 hitch weight not sure if that rating changes with the use of a Weight distribution/sway control hitch or not. That Zinger has a dry weight of around 3100 and around a 400lbs dry hitch weight.
It really sounds like you are over taxing that Santa Fe. As someone else had stated you can get some helper airbags in the rear suspension but you should really not use those to compensate for an already overloaded suspension (If that is indeed the case). I installed those on my Nissan Frontier when I was towing my old Hybrid Rig, it was VERY easy to install and made a world of difference when towing. The rear end rode perfectly level, I didn't get alot of bounce when hitting bumps and I felt it really helped me maintain speed when going up steep hills.
As someone else mentioned, single axle trailers are prone to a little more back and forth wiggle so you might be feeling the effects of that as well.
What type of tires do you have? If you have those larger marshmallow like truck tires, that will allow your vehicle to move back and forth a bit and you will absolutely feel that when towing. They do make stiffer tires rated to tow that help with that sway.
Do you have a brake controller? have you checked your configuration? when I got my new trailer apparently the controller was set much higher than it needed to be, so when I applied the brakes the trailer would brake MUCH harder than it had to and pull my SUV back. When releasing the brakes it would release very quickly and I would feel my SUV lurch forward a bit. So double check that.
PLEASE PLEASE Be very cautious about towing with a vehicle that is beyond it's capabilities, There are some real horror stories people have had told me, from not stopping in time to almost tipping over. I used to tow a 1800lbs Pop-up with a Kia sportage and I KNOW firsthand those Kia's/Hyundai's can tow more than they say they can but still, just be cautious.
If your searching for a comparable replacement tow vehicle, check into the Nissan Pathfinders or Frontiers, those things are just pure beasts when they tow! I used to tow a 24 foot Hybrid with a Frontier and now I tow a 30ft with an Armada and It blows my mind how much power they put out.