You don't need to use that agency for that authorisation number. All the consulate cares about is that you have a valid number to proceed with issuing the visa. All the immigration officer at the airport cares about is that you have a valid visa.
Also, you don't want to be applying for a visa in the lead up to or during Nowruz. The MFA staff stop working well before the official holiday period.
Separate the issues. Tell your travel agency that you will arrange your own visa, and instead spend your time worrying whether that agency can deliver the promised tour. Perhaps abandon the tour. Travel within Iran is cheap and easy to arrange, although you will compete with Iranians for hotel rooms and seats on transport during Nowruz.
Don't wait until March to get this process started. Apply for the authorisation number via another agency three to four months before arrival. The code is valid for one month. Depending on the issuing consulate, you can get the visa within one hour or one week. The visa itself is valid for entry for the three months following issue.

