The specs of the power supply that comes with a Galileo Gen 2 are 12V/1.25A, just as you pointed. I recommend you to use power supplies with a supply voltage of 12V and a current of 2A. You can keep using the 12V/1.25A power supply that you currently have, but if you use shields along with the Galileo board, the complete system might need more current and a 12V/2A power supply might be more appropriate in that case.
Regarding your case, I would suspect about the board. The issue might not be in the power supply, but in the board itself. The 12V/1.25A power supply should be fine, but if you want, you could get a 12V/2A power supply. Make sure the power supply is working as it should before powering the boards with it. If you are sure about the power supply, then the boards might be defective.