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If your state of residence is CA, all of your income is taxable in CA unless specifically excluded. When you work in one state and are a resident of another, you would normally be taxable on the income earned in the state you worked (WA). That income would also be taxable in your home state (CA). Tax laws allow for a credit for taxes paid to another state on your home state return. However, because there is no personal income tax in WA, there is no credit available on your CA return.
I can't speak to the CA back taxes owed as that is a legal versus tax matter.
No, if CA is your secondary residence, you would not be subject to tax on your WA income. However, if that was true, you should have notified CA of your change in residence.
CA could be classified as your tax home if you spend more time in CA than WA. Also, if you have a CA drivers license, CA plates on your vehicles, vote in CA, CA hunting or fishing licenses, etc there is the appearance that you are a CA resident.