According to your description, the internal voltage regulator is damaged. The USB power supply is connected after the voltage regulator, that's the reason why the board is still working when you power it through the USB port.
If you got the Edison recently, I highly recommend you to contact the place of purchase and explain the issue to them. Hopefully you are still under their warranty period. If you are out of their warranty period, you can submit a warranty ticket using the following form and the Warranty Team will contact you as soon as they receive the ticket: Intel Support.
Very helpful, Diego! Thank you!