I'm feeling conflicted because my driving instructor said that you could and maybe should use handbrake for additional barking power in an emergency, but car's manual says that it shouldn't be used and may be damaged if used while the car is moving..

Do not use the handbrake for emergencies. You don't have antilock brakes with the handbrake and it will probably throw you into an uncontrollable skid rather than slow you down.

How would it be damaged if used while the car is moving? It's just a cable that activates the rear brake caliper independently of the hydraulics.