Welcome to the expat forum, grace. Are you a qualified teacher or going to teach ESL? If you've just been offered a job, they are probably trying to fill an empty position right now, in a school year that is underway, so certainly wouldn't have time to get a proper visa.

Have you asked to talk to other teachers at the school? Do they have anyone there who did the same thing? They don't have to be South African, just anyone who came in on a tourist visa and got a work visa. You might be able to get some information on Dave's ESL Cafe. It's for ESL teachers, but the visa experience should be the same.

In some countries, people working on tourist visas, at least initially, is the norm. Your own embassy is never going to tell you to do anything other than follow the letter of the law, no matter what the reality is. In parts of China, the norm is for teachers to come in on a tourist visa and convert.

You could check the Mexican Embassy and Consulate web site to see if they mention conversion of visas.

Unfortunately, if you get in trouble, Mexico jails illegal workers, I believe.