You are a resident of VA and non resident of NJ. Complete the non resident return in TurboTax first.
You must file two state returns: a non-resident NJ return, and a resident VA return.
Your resident state of VA can tax all your income, regardless of its source.
NJ can tax you as a non-resident only on NJ-source income. Income is normally "sourced" where the work is actually performed. However NJ suspended this rule during Covid, allowing NJ employers to "source" income according to the employer's jurisdiction. This Covid-related suspension ended on October 1, 2021.
Thus your income prior to October 1st was sourced to NJ (your employer's jurisdiction); your income after that date is sourced to VA, where you actually perform the work. NJ cannot tax a VA resident on VA-source income. Therefore, on your non-resident NJ return, you would only report your income earned prior to October 1st.
You'll be able to claim a credit on your VA return for the taxes paid to NJ, so you won't be double-taxed. As @Bsch4477 pointed out, in TT you should complete the non-resident state return first, so that the program can correctly calculate and apply the credit.
Source: https://www.state.nj.us/treasury/taxation/covid19-payroll.shtml