How will my power/current draw change if I switch from 120 VAC to 12VDC?
Actually, it will draw 1/10 of the current at 1/10 voltage, producing 1/100 (1%) of the original power and heat!

This is because of Ohm's law: voltage = current * resistance (E=IR). Using algebra, I=E/R. Because the voltage, E is 1/10 while resistance, R is constant, I is 1/10.

Power = voltage * current. Because both voltage and current are 1/10 of the original value, power is 1/100 of the original value.

Your heater will produce 1% of the heat at 12V. Sorry.