Followup on this: I fixed the problem!
It needed a new front hub. After searching elsewhere I found the problem was a front ABS sensor, as Jerry suggested. It's a common problem for Explorers like the Gen1 ST. The symptom is that the brakes felt soft and pulsating while coming to a stop, at less than 3 MPH, especially with the wheel turned one way or the other. In my case it happened with the steering wheel turned to the left, and it turned out that it was the left hub that needed to be replaced, although I only figured that out after first replacing the other one. Basically what happens is that the ABS sensor wrongly indicates that the wheel is not turning, causing the ABS controller to pulsate the brake. It is a safety issue, although probably not one of life or death.
For anyone who wants to do this, it requires about the same skills as replacing brake pads and rotor, since you have to take those off anyway to get to the hub.
I used several project write-ups, including these three:
* http://www.explorerforum.com/forums/showthread.php?p=1499806 (very nice writeup, including lists of tools required)
* Project at mysporttrac.com on "brake job"
* Project at mysporttrac.com on "hub bearing replacement"
One thing not mentioned in these writeups: when you are all done, if you feel a shudder in the steering wheel, it's due to air in the power steering fluid, which happens when turn the steering wheel without the engine on -- that's something you're likely to do in order to reach the bolts that hold the hub on. You fix this problem by repeatedly turning the steering wheel until the air is gone - this is easiest to do while the truck is still on the stands.
I used the Timken SP450200 from Rock Auto for $100 each, plus $5 each shipping. I also needed a hub nut, $14 each from the Ford dealer. My total cost was about $240. The hub includes the ABS sensor. So essentially I bought a whole hub just to get the new sensor that comes with it. It includes an electrical cable for the sensor, which connects to a mating connector behind the front bumper.
For one of the two Timken units that I bought, the ABS connector would not lock onto the mating connector, so I used some blue loktite where the two plastic connector housings touch -- I've never done that before, so I hope it works.