Don't adjust the steering box worm gear (the screw you are referring to on the top.) You are asking for trouble. If you are convinced the steering box is worn out (IMO not likely) then replace it with a rebuilt one for about 150 - 200. More likely is that the slack in your steering is a combination of things, esp. the steering shaft which no one ever replaces.
I don't tow a lot so I don't know this for sure, but maybe your tongue weight is too much? WOuldn't that cause the front end to be unloaded and wouldmake a shimmy happen? Again I don't tow much, obviously less than you, so I don't know for sure.
Obviously make sure the tie rod ends and drag link ends are good.
As far as a steering stabilizer goes, I would say you'd probably eb happy running a dual stabilizer like this one
http://www.4wheelparts.com/product2....HHG6MTDPJXFWVF
The other thing to check is the condition of your kingpins. For the upper, the nylon bushing is something that wears out on occasion and is very easy to replace. On the lower, the bearing, if worn, can cause excessive play. You can get a complete kit from parts mike
http://www.partsmikeparts.com/dana_6...ebuilt_kit.htm
Everything except for the kinpin itself, the black tapered metal piece in the upper right hand corner under the nylon bushing, is very easy to replace.
-- Mike