In cases like this, the first thing to do is replace the battery and see if the problems persist. You already did that, and they did, so then the next step would be to replace the motherboard as there seems to be something wrong with the charging logic. There's also a small chance that your AC adapter is damaged and not supplying sufficient power to run the unit and charge at the same time. You can test this by shutting the unit off, leaving it all plugged in, and coming back in 2-3 hours. The battery should be either fully charged or very close by that time, and so if it is not significantly further along in the charging process, you may want to find someone with a multimeter to test your AC adapter and make sure it's putting out what it says it should on the label. If that tests out fine, it would be time for a new motherboard or some kind of external charging solution.