My iPhone 5 dies when not plugged in.

Question

I got a new iPhone 5 from the Apple Store a little less than a month ago. I haven't dropped it, gotten it wet, scratched it or anything since the time I recieved it, and it always has a case. My phone died unexpectantly at 80% and when I plugged it into the charger it took over an hour to turn back on, when it did, it still had 80%. I unplugged it when it got to 100% and it lasted for less than ten minutes until it died again. I plugged it back in and it took relatively thirty minutes to turn back on, when it did, it had 100%. I tried unplugging it again, this time it lasted until 86% when it died. This continued every time it charged. I tried using three different USB cables, tried plugging it into the wall, my laptop, and even a friend's portable charger, and the problem still happened. It was slowly getting worse (dying as soon as it was unplugged with 100% charge) I went to the apple store and they ran diagnostics on it, it came back with nothing wrong. Battery was fine. The man told me to go home, back up my phone, and restore it to original settings and it should be fine. He expressed to NOT RESTORE MY PHONE TO THE BACK UP. I spent roughly two hours restoring my phone, setting it up as a new phone, and downloading apps, adding email, music, books, etc. As soon as I unplugged it from restoring it, it died again. I am getting angry with the iPhone 5, as my old one died and wouldn't turn back on, and it is the only phone they will give me as a replacement. It took three hours out of my time to go to the apple store and be told something that doesn't even work and would take two hours to accomplish. I am currently using my laptop for iMessages and when in need of my phone having to be with a charger and and be around an outlet at all times. Is this problem fixable without having to go back to the Apple Store or taking apart my phone? My phone is not jailbroken, nor have I downloaded anything onto it besides syncing music from my iTunes Library. It is up to the latest software update.
It may not be relevant but the service bar will always say 'searching' even though I have service and can send messages.

Answer 1

Basics from the manual are restart, reset, restore from backup, restore as NEW device.  If those don't solve your problems, you'll need to go to Apple for evaluation.