The 7805 regulator you are using appears to be in a TO-220 case. That device, with no heatsink (as in your breadboard) has a maximum power dissipation of about 2 watts. With 12 volts input it must drop 7 volts across the regulator. You are drawing at least 250 ma. This requires the 7805 to dissipate 1.75 watts. I would expect the 7805 to get too hot to touch. If you are actually drawing 300 ma, the 7805 will be over the maximum and may shut down.
You might put a 10 ohm, 10 watt resistor between the battery and the regulator input. This will allow much of the voltage drop to be across the resistor instead of the regulator and it will run much cooler.
Mike