I now present myself for the "Bonehead of the Month" award:
Well, I checked the bushings. Everything looks fine; the body bushings are (almost) just as perfect as the day they were new (surprisingly, they're barely even cracked slightly). Control arm bushings are a little more worn, but still perfectly serviceable.
I didn't get a chance to jack the car up and check the U-joints... but when I did open my trunk to get my hydraulic jack, I discovered that my full size spare, sitting in "The Pit" (where the spare tire is supposed to go) was standing up straight. I could roll it forward and back with my pinkie. Which, at least, explains the thump and gets me my newest bonehead award: when I hit the brakes, the spare rolls forward as much as it can in "the pit". Once I've come to a stop, its forward momentum stops and it plops back, hitting the rear wall of "the pit" and causing a thump.
Crisis averted.
I also checked the steering as per the instructions of 85 Cutlass Brougham. There is a little bit of play, but not more than I would expect. I'm going to replace the rag joint as soon as I get a nice day to play around on.
Thanks for the help, guys!
AJ