1. Yes, do send the resignation by post as well e-mail to the official e-mail address of the appointing authority only if you are a temporary/ad hoc employee. Meres sending to resignation to Principal is not enough.
2. You have to do the same to the authority which issued you the appointment letter with copy to the Principal.
3. However do check with your new employer to find put whether without a valid release letter they can appoint you in a new post .
4. Since you appear to be a full time employee I do not think mere sending of resignation letter will do. Absence of a valid release letter may jeopardise your new employment.
5. However nothing to worry as in that case a filing of writ petition would solve your problem.
Feel free to contact in case of further need.