Do higher wattage light bulbs draw more amps?

Yes, and it's proportional.

V * I = P

Where V is the voltage in Volts, I is the current in Amperes, and P is the power in Watts.

So we get:

I = P / V

For example, with a 240 Volt supply, a 12 Watt lamp would draw:

12/240 0.05 Amperes, or 50 milliamps