You can come in China on tourist visa, but you can't work (and receive pay) with L visa. It's illegal to work with L in China.

It's much easier for School to get FT on L visa.

Most likely, they promised you different visa later, butT....once you're in China on L and working, your employer has handle over you.

You can't complain too much, and request promised things (salary can be lower, housing can be shitty, and so on).

Contract you'll sign is invalid, and you can't expect help from FEBureau.

School is probably not authorized to hire FT, in other words, you'll probably never get Z, but you'll be stuck with F visa.

If School will sponsor Z visa later, you'll be required to return to your home country for Z pick-up!

Get a Z visa and contract first, don't leave home without it.

The only case I've heard about someone successfully going to China and then converting their L to Z was a guy working at a Fortune 500 company who had everything in his contract signed in Germany and then just went on a month long vacation to China before settling down.

So except a few select case here and there I wouldn't trust an employer, and especially a Chinese school to actually help you change to a Z visa. I wouldn't travel what I assume is half way across the world for the chance of working illegaly. 

Whatever you do, get the Z visa first. Don't believe a word they say about converting L visa to Z visa. If you get caught working on a L visa you can kiss China goodbye. They will fine you, put you in detention, deport you and ban you from entering China for a number of years. It's not worth the risk. 

What others say... don't get yourself in a world of trouble for not taking good advice when it was offered.

It's perfectly fine to come and work on an L visa, if you want to break the law and be deported and banned for several years.

Otherwise, ditto to what everyone else said about the "Z" visa.