OP, just fix the swaying....if it’s actually swaying and not just how a trailer generally behaves. Sounds like your experience with towing is very limited and with that your expectations may be out of line a bit.
Does it bottom out your air suspension? If not, it ain’t overloading the back of your rig.
Have you hooked up the wdh and sway? Your post is unclear if your issue is just a “before” issue or an “after” issue as well.
Yes, you’re at the upper end of the “ratings” of your Escalade, mostly because the Esc is a very soft plush suspension vehicle. That said, it absolutely has the length, mass and power to tow that trailer competently, regardless of what the rvnet weight police say.
Provide some more info if you wish, and then wade through the plethora of “you need a HD pickup” responses, if you need more info or suggestions.