Yes, you are a nonresident, and yes, you may file Married Filing Separately in New Jersey. This may be done in TurboTax, but it is much better to do so with desktop software. Please note the following FAQ that describes the process: https://ttlc.intuit.com/replies/3301995
You will use the status of Married Filing Separately since your Federal Return is Married Filing Joint. You are not allowed to use the Single filing status. Since you do need to create a mock, or an "As-if" return, the New Jersey return will have to be mailed in.
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