No. Nothing is double taxed because there is no overlap of income. Part-year residents only pay tax on income earned while they live or work in a state.
On your part-year California return, report the income you earned from January to March. Since your employer withheld tax for the entire year you will have a big refund. You are being taxed on three months income but had withholding for 12 months.
Report the rest of your income to New Jersey. You will owe nine months of tax since your employer did not withhold anything for New Jersey.
NJ like federal extended its filing deadline to May 17. You should be able to file CA and get your refund and use that to pay your NJ tax.
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