Moving from a state without a personal income tax to a state with a personal income tax

I have kind of an odd question that has been difficult to find an answer on my own. I am potentially moving from the state of Washington (no personal income tax) to California (has a personal income tax). I am a 4th year graduate student at a university in the state of Washington and have established residency for in-state tuition purposes. Lastly, for my employment I am teaching courses for the University remotely from California.

 

As the state of Washington does not have a personal income tax (i.e., no potential for me to get double taxed from both states), my question is if the university sends me a W2 at the end of the year referencing a Washington address for myself, can I, on my own, file my taxes as a resident of California, effectively paying the CA income tax? Or do I need to notify my employer to fix the state on my W2, and then file my taxes with the corrected W2?

 

Thank you!