Go to your System Settings on the phone. Go to Cellular. Scroll down. It should say “Use Cellular Data for:” Apparently iOS6 defaults to “iCloud Documents on.” That means it was syncing everything on iCloud while on cellular rather than WiFi. I can’t think of what it was syncing, but it might have been Downcast data. In Downcast I’d told it to sync with my iPad. But there wasn’t any setting to specify only on WiFi. So I wonder if Downcast was syncing all the podcasts I’d downloaded before leaving. If so, then that might explain a lot.
To jump ahead of myself, the solution once again was to delete corrupt Safari bookmarks, but what’s easy on a Mac is often difficult or even impossible in iOS. I’ll share my unsuccessful intermediate attempts, and if you’re experiencing similar battery life problems, I encourage you to try the less-destructive approaches before taking the eventual tack I did.