In your self-described circumstances, yes. Exactly.
Your legal residence (actually called "domicile" in the tax law) changes whenever you physically pack up and move to a new state (or country), without the intention of returning thereafter.
For instance, if you lived in Washington state, and then went to California to attend college (but planned to return to Washington state after you graduated), then your residency hasn't changed. Similarly, if you go to California to work on a longer job project, but maintain an apartment in Washington, you haven't abandoned your Washington residency (yet) and would be considered a California non-resident for tax purposes.
However, if you've left Washington state for good, and relocated to California, exchange driver's licenses, registering to vote in California, etc., then you've become a California resident. Because in your case that happened around March 1st, for the 2016 tax year, you are a part-year Washington resident (January 1, 2016 to March 1, 2016), and a part-year California resident (March 1, 2016 to December 31, 2016). Next year, if you continue to live in California all year, then you'll be a full-year resident for taxes in 2017.
So, you'll have to file a federal return for 2016 (because federal is federal wherever you go), and a part-year California tax return, to report any income earned or received after your date of relocation and residency change. Since Washington state has no personal income tax (lucky you!), you won't have to worry about filing any tax return there. Although if you did have to file a tax return for a state with a personal income tax system, like Oregon, then that would be a part-year state return also.
Thank you for your question, and we hope this helps to clarify the multi-state issue.