Charge port on the phone may need replacing
The charge port on the phone will not accept the charge unless you "wiggle it" and it beeps to inform you the phone is charging.
Does sound like an issue with the charging port. Have you tried a new charger and cleaned the charging port on the device to make sure? The S3's are known to have charging port issues so this is not surprising, but unfortunately with this model it does require soldering and not an easy DIY job.