I am not new to teaching ESL or China. However, I am going to a new school in China. I am female, single, and older (not so valued)

Fair enough.

I have experienced before where female chinese staff who process the visas, put up a non-verbal block in order to prevent the next year's visa not being issued.

Prevent ........ not being issued = ......... is issued! No problem, so no question. However, your English ain't good!

My question is how do I manage the politics with female chinese staff?

Your question is irrelevant!