Figured it out. Shortyz creates a directory on the SD card called 'crosswords'.
('Duh', right? I did have another program on the device that's actually called 'Crosswords', so maybe I'm not a complete idiot .. just a partial one).
Anyway, dropping my '.puz' files in that directory makes them show up in Shortyz. There's also a sub-directory called 'archives' that - you guessed it - makes them show up in the 'archives' list. Interestingly, I copied a .puz from the directory I had been using without copying the .shortyz file (which I assume stores progress), and Shortyz seems to find the .shortyz file, and show my progress.
It would be nice if I could make my own directories to organize a big pile of puzzles (and have Shortyz recognize the dirs), and/or if I could browse for other directories from within the program - but that's not within the scope of this question. Maybe I'll submit a feature request or (gasp) submit a patch (yay, FOSS).