The most recent version of Palm Desktop has three sets of address
fields instead of one: "Work", "Home", and "Other".  Upon upgrading,
all my existing addresses were unceremoniously dumped into "Work."

Since then, I've made corrections, moving home addresses into "Home"
and the like.  The problem: I can see the multiple addresses within
Palm Desktop, but only "Work" is accessible from the PDA.  How can I
access both "Home" and "Work" (and "Other" if I use it) addresses on
the PDA?

You might be interested in a third-party solution.

Chapura's KeySuite is a good one, but is a little pricy.
http://www.chapura.com/keysuite.php