URGENT: Visa / Immigration info (Mexico City)

Hi,
I'm from Australia and living in Mexico City. I have been offered employment but the visa papers haven't processed in time and my tourist visa will run out on June 2nd.
I was wondering does anyone know if you can extend the tourist visa with a letter from the lawyer and proof of papers having been applied for? Or do I definitely have to leave the country to refresh the dates.

Also what happens if I overstay the tourist visa and leave the country once the papers are ready, do I pay a fine for the days I overstayed and will I have dramas coming back into the country after a few days with the new working visa?

Any help or advice is much appreciated as I'm stressing over this. It doesn't make sense financially and timely to leave the country only to have to do it Protected content later for the visa lodgement.

Kind regards

Brendan