OK. I've got some pretty clear results.
Everything there is pretty dry, but that's sort of a result of driving in the desert then power washing off all the dust. However, there may be a little grease at the ends of the u-joint caps.
Also, while I was feeling around the caps, I was able to wiggle the axle shaft a bit and feel some movement on the passenger-side u-joint. That's a pretty clear indication that there's a problem.
I tried that, but it didn't seem to do anything. Maybe the u-joint is far enough gone that a little spray lube isn't going to help.
Following
@qslim's mention of the ChassisEars, I wrapped a Bluetooth headset in tissues to cut out some of the surrounding noise and duct taped it to the diff cover. Then used it to call myself on another phone. Made a couple three-point turns and could hear the popping out the window but not through the phone. Moved the headset to the steering knuckle and could hear the popping clearly through the phone.
So, we have a winner. Bad passenger-side u-joint. I'm ordering Spicer 5-760X replacements.
I dug up some of the online write-ups, and I guess I'm willing to go ahead and tackle this job on my own, even though I usually make a hash of anything that requires a press. Looks like Stu Olson has some good tips on removing and installing u-joints with a ball joint tool:
https://www.stu-offroad.com/axle/ujoint1/u-joint-1.htm
Thanks everyone for the tips!