New iPhone 5 Battery Problem

MY phone was working without any problems until 2 days prior to my heading off on a 2 week trip overseas. I contacted AT&T to get my phone unlocked. This process involved completely erasing the phone and restoring it from a backup (via iTunes). BIG MISTAKE. Right after doing that, I started having a problem keeping the phone going for more than a couple of hours after a full charge, instead of lasting all day as it had previously. I would get no low battery warning; it would just shut down and give me the low battery pic if I tried to reboot it. I'd plug it in and the indicator would show 89-100%, which seemed to be a more realistic indication of the battery's actual state, given the small amount of usage. After 15-30 min of charging, the same thing would happen.

In its original, functioning state, the phone was running iOS 6.x, since I hated the upgrade's look & feel. I think the restore left it at that version, although I had a suspicion that it was an earlier version than originally, but I'm not sure. The battery had been replaced under the recall approx 6-12 months ago. These are the steps I have taken since, being out of the country with no Apple Store available. Everything has seemed to make the matter worse, and worse:

FIrst, I upgraded to iOS 8.3, thinking it was the old iOS problem

Second, I upgraded to iOS 9.0.2.

Then, after backing up to iCloud, I erased the phone and restored from that backup I'd just made.

Today, after a full night's charge, I lowered the backlighting to 1/4, and could barely keep the phone running 5 min. After recharging again, it wouldn't run long enough for me to sign in.

Throughout all all of this time, the phone was using either cellular data or wifi and Bluetooth was on, generally connected to my headset.

Any help will be greatly appreciated as I will be abroad for at least another week. Thanks.