Shaky towing experience, please advise

Easy, cheap things first. For your next trip. Front tires to normal pressure, rear tires to sidewall max, trailer tires to sidewall max. Load all stuff forward of axles, don't throw stuff back in the bunk area. And try to keep waste tanks as empty as possible. Make this your baseline for any further changes. If you still feel the loose greasy steering check the level of the trailer. Measure ahead and behind the axles on flat ground. This will tell you if you need to lower the hitch head 1 hole. And putting 5 gallons of water in the fresh tank to increase tongue weight may help. As someone else mentioned, do 1 thing at a time. I also think that your brand new tires may be some of your issue. This will become a non-issue as you drive the truck and wear them in. You have very near the empty weight on your steer axle with the spring bars pulled up, I don't think you need more.