Sounds like the problem is the battery itself. Definitely see if you can get ATT to replace that with a new one. Also you may want to replace the charger & cable as well. Having the battery level drop while it's still plugged in seems very unusual to me. I've used many devices while they are charging and have never seen that. I ordered an additional USB cable from Amazon for about five bucks. One with the charger adapter is less than $10. Amazon.com: OEM HTC USB Travel Charger Adapter U250 / CNR6300 / 79H00095-14M: Cell Phones & Accessories
The seller I got mine from is AccessoryOne and it's OEM and haven't had any issues with it whatsoever. Others have reported that some non-OEM cables have caused problems. And I don't think ATT sells OEM chargers/cables. Even if they did it would probably cost 3x as much.
And switch up where you plug in the phone. Maybe the outlet where you plug it in is loose and the plug falls out just enough to break connection. I had a CO monitor plugged into a loose outlet before and every now and then it would drop out just enough to trigger the alarm. Usually in the middle of the night. Now that's not fun!