Oh, hey--I guess I left you all hanging and forgot to report... the rest of the story. </Paul Harvey>
For the record, my originals were Power King Towmax tires, C1 load rating, size 205/75-15. As I looked around the web, I saw a great number of horror stories about how awful these cheap Chinese tires are. My experience was exactly the opposite (until now, anyway). For five years, they treated me great. Was I unusually lucky? Maybe it's because I would check and inflate them to 50 psi before every use? Perhaps it's because I would never exceed 70 MPH while towing (except for those rare occasions when I had to keep up with Brett as he blasted across I-80 in Nevada)? I have no idea.
I took Jeremy's advice and ordered a set of load range D1 tires in my same size; I ended up choosing the Carlisle Radial Trail HD. I found a number of good reports about them in various forums and such. Icing on the cake, there was a sale at Discount Tire which got me $100 off any set of 4 tires that their site rates Best, which applied to these. They were shipped to my home, at which point I slowly towed the trailer across town to have the local DT install them for me. Once they were on, the trailer immediately went back to its former smooth operation.
I am planning a quick Moab trip in a few weeks, so I decided a shakedown run was in order. Last Tuesday I loaded up and towed my Jeep out to 5MP to run the snakes with a co-worker. Everything went perfectly, so I am confident this was my entire problem. I feel ready to head to Moab.