If you used 10mA as current value for the white LEDs, it's probably not enough. Try inputting 25-50mA into the website LED calc.
3AA batteries may work, but they don't have to - it all depends on the LED forward voltage: if it's more than 1.5V then voltage from 3AA's in series won't be enough - you'll be better off with a 9V battery, 4 AA's in series, or if you want to use 3AA's connected in series, then connect the leds in parallel, like this:
180R means 180 Ohms value - this will give you 52mA forward current. But you can as well use bit lower resistors for brighter LEDs - down to about 90 Ohms, which will give you 50mA forward current. But i'd recommend starting with the higher value
Also, reversing circuit polarity shouldn't damage the LEDs, and polarity of the resistor is irrelevant.