All's better after new battery
I just wanted to post a followup to my issue. After the shutdown mod and battery calibration didn't help, I bought an Anker replacement battery from Amazon. Since my wife had the same problem, I bought a two pack for $28. I charged the battery overnight and then in the morning hit the battery calibration button.
I noticed a difference right away - I was only down 7% after a few hours at work. My goal was to let the battery run down until the phone shut off, which happened at about the 30 hour mark with my playing HD youtube videos for the last hour so it would die. The battery life indicator actually counted all the way down to 1% before shutting off, which is a stark contrast to the 30% that had been occurring with the old battery.
With the new battery, I've also used the camera when it was under 30% with no problems.
My recommendation for anyone that has similar problems is to get a new battery. They are a hell of a lot cheaper than a new phone, so it's worth a shot.