Some follow-up. I was going to return the SMX-10, but there was no way I would get a replacement in time, even if I knew what to get, so I held on to it. First, I tried it outdoors in a very quiet environment. I recorded some birds, a woodpecker going at a tree, and nothing at all - just to see if there was noise. It seemed OK. Rather than "much worse than the built-in mic", it seemed better - nicer tones, maybe a little more noise.
So I took it along to this event and recorded about 5 minutes of a speech before an audience. The audio wasn't bad. The mic picked up a lot more side noise than I would have liked, from people around me before they quieted down - "shotgun" is a nice term, but of course it isn't really like that. Further on, I got some weird interference sounds which I think I recognize as a GSM phone talking to a tower - I've heard this if I have an AM or shortwave radio on near my cell phone (on but not in use), but I don't know why the camera or mic would pick it up (it isn't audible). After that, the audio was fine.
So now I'm not sure what to do. I'm thinking of called Azden to ask about the SMX-20, which is a newer model that says it was "designed for DSLRs". The specs don't look much different to me, and it is so new it doesn't have reviews, but I would like to hear what they say about it.