If that stepper is just turning a leadscrew, you could use almost any little stepper to do that. Plastic does not conduct heat very well.
Learn to fabricate aluminum. You could make a plate to go between the plastic mount and the motor and fold the edges up and around the motor to conduct the heat away.
But, I really think you are running the motor with too much current.
Does your motor need to hold the leadscrew in position, or will your pen stay where you put it?
Paul