There seems to be two questions here. Firstly can you get a z-visa and secondly can you find a job in the first place. I'll take them in reverse order.

I'm sure that you will be able to find a job. There are many Filipino teachers working all over china, teaching in various different types of schools. However, by in large as a non-white teacher from what is considered a non-English speaking country you wouldn't have quite the range of options available to you. Your best bet is probably finding work in a third tier city or large town which struggles to attract teachers from the so called Big 5 English speaking countries. These jobs aren't always easy to find without contacting a recruiter but they are definitely out there.

Whether or not as a Filipino you will be able to get a z-visa/residence permit is a different question. As one of the other posters has suggested it'll depend on which city/province you are in and how they choose to interrupt the conditions for issuing the relevant paperwork. An alternate would be to just arrive on a long term tourist visa (l-visa) which you should have no problem acquiring as you are married to a local. This would make working illegal but it is far from uncommon to find foreigners of all nationalities and levels of experience working in China on L-visas. Again, if you can be more specific about the area you are looking to work in I'm sure someone will know firstly if its likely you'll be able to get a z-visa and secondly whether the authorities are strict on foreigners working without all the official documents.