Windup, I'm primarily using my iPod (nano), so iTunes being buggy with podcasts is unfortunate. (What, am I going to go back to listening to music?)
On my iPhone, the podcast app seems to be working, so I might try to migrate my listening to that; or possibly try some other OS app like Downcast if it can sync with an old-school iPod.
Still, this problem has only cropped up in the last few weeks--around the time I updated iTunes (which is up to date)--so I'm hoping that Apple might fix it.