I'd start with a "cheap" investment with something like these.
https://www.amazon.com/Camco-Olympia.../dp/B000760FWU A pair (all you need) is $21 or you can get 4 for $ 33.
Use two at the extreme rear of the I-beam rails (not on the thin bumper extensions) to "tighten up the rear sway". You can "more easily set them" by leveling your trailer, then using the electric tongue jack to lower the tongue 1", set the rear jacks in place, raise the tongue jack back to level to tighten the screw jacks in place. Easy to do and only 2 extra trips from rear to front of the trailer.
If they work, then start formulating your "permanent solution" which may be as simple as to continue using the screw jacks...