You should register your new phone in micorosft cloud in order for it to work.
to do this , point to the old phone MS authenticator and choose backup. this will lead you to backuo all the data to another MS cloud account.
then from new phone restore what you backedup.
If you can not get your old phone to work. speak with the school help desk so they can reset the MFA for you. This is normal user call for new phones.
Hope this will help.