I feel, in case you settle all dues from school, it means you accept the termination and there the matter ends. if you decide to close the matter and find another job, this may be done along with taking service certificate etc. But in case you decide to challenge the termination, this may not be done.
Teacher is not a workman as decided by High Court of Kerala in Venkataraman vs Labour Court (1982 I LLJ 454 KER). In Ms. Sundarambal vs Govt of Goa, Daman and Dieu & others (1989 I LLJ 61 SC), Supreme Court also held that teacher is not a workman, though School is an industry. Hence, remedies under Industrial Disputes Act 1947 (intervention of Labour Officer, Labour Court etc.) may not come to your help.
Invoking appellate provisions of your service rules (under the state education rules) may be an option as pointed out by Shri Umakanthan, Addl LC (Rtd) above. There is a need to find out which rules will apply in your case (may depend on whether school is aided, unaided, run by any society etc.), and only an advocate practicing in the area can help in these matters clearly.