I would like to inquire about the best provinces and cities to work in in China, if you are
over the age of 60 years old.

As of last year, Chinese embassy/consulates will not provide a Z visa for anyone over the age of 59. You cannot BEGIN to teach English legally in China at 60 or over. You can renew your FEC at existing and new employers if you are already teaching in China until the age of 66.

I know of teachers in their 60s teaching in Fujian, Guangxi and Hainan. I'm sure that there are other provinces.

If you are a professor at a Western University, I'm aware of instances where universities in China provide business visas, offer travel reimbursement and housing for lectures on a short-term basis, perhaps a semester or two. Although a salary for that activity would be illegal, I suspect that there are 'honorariums' or some other financial incentives provided.