This is one of the most frequently discussed problems with our trucks. Search on "steering wander" or similar words, and you should find tons of reading. There is a techical service bulletin that shows how to make two adjustments to your box; one is easy, one is a little more involved. But if it is really bad, a good dealer will install a new steering gear box (mine did - Hagan's Dodge, Morrilton, AR - THANKS!)
Other issues that have played a role or been blamed for steering wander include track bars, setting up the alignment at one end of the factory-allowed specification range, adjusting tire pressures, and replacing the steering damper / stabilizer with a better aftermarket device such as one made by Rancho. Folks with good dealers have gotten some improvement; others have fought and fought. But you are not the first to deal with the issue, so happy reading, and if you have trouble finding the threads e-mail me and I'll see if I can locate some of them for you.
I don't recall seeing anyone who had solved it by going to a '97 gearbox; sounds creative; is your dealer going to foot the bill and still guarantee everything, or is he just sending you away to get someone else to do this? Who knows, maybe it will work!
Good luck, and welcome to the TDR!
P. S. - I almost forgot, a tip from previous threads: when discussing with your dealer, use the key words "excessive play", "excessive steering wheel play", or similar. That is what keys him to look at the right techical service bulletin and gets you started down the right path.