When I used airbags with my F150 and TT I loaded the truck 1st, then aired up the bags to get the rear of the truck back to original. Then hitched up the TT and set the WD bars. Rode like a Caddy.
JMO and some may differ, but why not try and air up the truck to where you want it. Hitch up the TT, but don't crank the heck out of the WD bars. Just put about half the needed tension on them. The HA/PP type hitches don't rely on friction for sway control purposes. If you can air up the truck enough and after hitching have the front fender height returned to at least 50% of the difference then it should tow fine. My thoughts are that you got too much upward push from the bags and WD bars being cranked.
Also is the TT level? And your TW should be around 1400lbs. Anything less then you will get that porposing.