ATmega168 circuit not working without Arduino regulator?
The L293 and most variations are 4.5v minimum VCC. That is why you are not getting the Stepper motors to run.

The TS1084 is a Linear Regulator with MAX input of 12v. You are wasting (VIN - VOUT) * IOUT in heat. That's (12v - 3.3v) or (8.7v) * IOut. Even if just half an amp, that's 8.7v * 0.5A or 4.35 Watts! of energy into heat.

Tj = (Pd · θJa) + Ta
( 4.3W * 80°c/w ) + 21°c = 369°C Junction Temperature for the T0-220!
It's operating range is MAX TJ of 125°C. Of course it would go into thermal shutdown.

You need a switching regulator, because any reasonable amount of current from a 12v IN to a 3.3v out linear regulator, will cook it.

Also, you don't list the mosfet used, so we can't check it. I assume it also is limited to a higher voltage range. Same goes for the stepper motors. 3.3v might not be enough if that is what you are running them at.