Here is a hot wire foam cutter that I recently built. Most of the stuff like the wire, MDF, 1 X 2s, 1/4 -20 all thread, nuts and washers I had in the garage from previous projects. I spent about $40 on the dimmer switch, 120v - 12v (3amp) transformer, clamps, and aluminum angle. The cord was salvaged from an old hot glue gun.
It works really well, though it does take some technique; slow steady movement and dialing in the right temp. Templates or fences are necessary for straight cuts. The switch is on/off and the dimmer controls voltage thus temperature. I might add an LED as a visual indication that it is on and/or a momentary foot switch for additional safety.