OK, here's what I tried. I went to Best Buy and put my battery in their phone for an hour. The screen was shut off, and nobody did anything with that phone. After an hour, it dropped from 52% to 50%. That's good.
Then, we put it in my phone, turned off the screen, and I waited in the store for 30 minutes. It went from 50% to 38%.
So, it seems like the phone was killing battery life for some reason.