Electrical – use dimmer voltage regulator for single phase multi speed AC motor

Do not put the whole shebang in series. Instead deploy your circuit with each battery individually connected to a motor and then through a N-FET to GND. PWM Drive on the N-FET gates will allow you to control the speed of the motors. Use two PWM signals to permit individual control of each motor's speed.

Selection of an N-FET type will take some careful work. Since your motor can draw 27A at normal load (133A at stall) you will need some beefy parts. If you design the circuit around nominal conditions plus a guard band you may choose a 70 or 80A N-FET. If so then you will have to design the circuit with some intrinsic current limiting or add some current overload sensing that couples back into the FET gates to shut them down when the current starts to get to high. Further details regarding FET selection are stymied by lack data regarding what you want to control the gates of the FET with.

Keep in mind that this configuration causes you to only be able to control the motor in one direction of rotation. If you require bi-directional control then you will need an H-Bridge for each motor that consists of four power FETs per bridge.

An 82 ohm resistor would dissipate about 175 watts; indeed, a 15 watt bulb may dissipate that much power briefly. Something the size of a 15-watt bulb that dissipates 175 watts is going to get very hot, however, and when tungsten gets hot its resistance increases by an order of magnitude. You may have been expecting your 1K resistor to dissipate about 1.5 watts, but in practice it would have dissipated about 17.5 initially. Most resistors don't nicely increase their resistance with temperature the way a lightbulb filament does, however, so your resistor may have continued to dissipate that much power until it melted, whereupon its resistance might have actually decreased, causing it to try to dissipate even more power, until things went downhill very severely.

I would suggest that you might consider a Christmas decoration that's designed for a C7 bulb; such bulbs draw about 1 Watt. If you wanted to use a resistor that drew less power, two 33K resistors half-watt resistors in series would probably be reasonably safe if they were somewhat ventilated (the resistors should dissipate less than 1/8 watt each, but half-watt resistors should be used so that in case one fails shorted the other would be able to handle the 0.44-watt fault power) but might not let through enough power to operate the touch switch (if it does work, keep adding 33K resistors until it doesn't work reliably, and then remove one). The fact that incandescent bulbs have a much lower resistance when cold than when heated allows a much smaller energy-wasting element to be used for a given application than would otherwise be necessary.