My trip report:
We made it from TX to FL with zero issues. I found a weight distributing hitch at Discount Tire and ended up buying the Blue Ox Sway Pro. Unfortunately the Uhaul trailer is not compatible with WDHs as it has an additional braking system attached to the trailer A-Frame that prevents the bars from being attached. The people at the Uhaul center were no help as they had never even heard of a WDH and it was a corporate center, not just a retailer who does Uhaul on the side. I ended up still using the hitch without the bars attached and it still towed just fine; the Platty pulled it like there was nothing back there. My wife had no issues with driving it either and she's never driven a truck with a trailer.
I know I probably exceeded my payload but like I said she drove like a dream and there wasn't much drop after I attached the trailer. The pics above are after we reached FL right before I took the car off and returned the trailer. The next time I have to tow a car I will either buy a trailer or ensure the rental A-frame is compatible with a WDH.
The pics aren't the greatest as they are from my phone. Thanks to all for the help/advice.