Yes, you should file state returns for both CA and NC.
When you have multiple state returns, you will normally enter the earlier part-year resident or nonresident state return first, then enter the state where you resided at the end of the year, in order to get the correct results. You are taxed by the state where you are a Resident (if that state has an income tax) on all of your income. As a Nonresident, you should only be taxed on income that is sourced in that state. The state where you are currently a resident may be able to give you a credit for taxes you paid to other states.
If you changed your permanent family residence to North Carolina, you would file that state's return last. However, you stated that your investment accounts are located at your "home address" in California. If you remained a California resident all year and just moved temporarily to North Carolina for a job, then you would file the North Carolina return first and the California return last.
Please see this TurboTax tips article for more information on filing returns with multiple states.
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