Yes, it is time to go to bed.
Connect VCC to 5V without a series resistor.
Series resistors on the Logic lines would be 2k2 or so.
It looks as if the backlight LED has a 10R resistor and is separate from the regulator on the pcb. This is probably for 3.3V and the raw backlight runs at about 3.0V. You would need an "extra" 82R if the LED is running from 5V.
Where did you get 300mA? Something must be seriously broken. e.g. a blown chip.David.