The issue turned out to be a Word setting that got unchecked when my workstation went through some recent upgrades. Specifically: File | Options | Advanced | Use fonts that are stored on the printer.
This is such a specialized application ("printing" to a fax server) using an old HP LaserJet driver that using another printer wouldn't have let me zero in on the problem. I remembered this morning that we'd had a similar issue a year or so ago and I dusted off my notes from back then.
Thanks to @MotoX80 and @Ian Xue for helping me eliminate the credential hypothesis.
Christian