This was the problem, though I don't know how I ended up in the situation I was in, as I was stuck on v1.0.301. When I checked the Google Play Store after seeing your comment, there was no update available, and the app page still had v1.0.301 as the latest version. I had no reason to suspect that my app was out-of-date until your post.
For posterity, I also had to join and then leave the beta program before v1.0.309 became available.