Update: After several weeks of trying things, here is what I found...
1st idea was to turn off 4G in hopes that the phone would not drain the battery as it was charging. I didn't have high hopes for this but it was the easiest thing to try. Worked for a few days and then the issue happened again.
In the meantime I installed the Battery Monitor Widget to see what exactly was going on. What appears to happen is that the battery attempts to charge but does so very slowly. After about an hour of only going from 62% to 69%, the battery all of a sudden jumps to 100%. It must be something that is programmed into the phone. Maybe they figure if you see it is at 100% you will unplug it and when when you plug it back in it will be on a better charger.
2nd idea was a better charge cable. I picked up a 3ft shielded cable from Monoprice and have not had the false charge issue since I have been using it. It provides a noticably faster charge than the cable included with the phone.
Problem is solved for now.