Yes, you will file two separate tax returns, one for each state. For this year (2019) you will have to file part-year resident returns for both New York and New Jersey. Next year (2020), if you live in New Jersey all year and work in New York for part or all of the year, you will file a New York nonresident return and a New Jersey resident return.
The income you earn for working in New York is subject to New York tax. All the income that you earn while living in New Jersey is subject to New Jersey tax, no matter where the income is from. However, you will get a credit on your New Jersey resident (or part-year resident) return for part or all of the tax that you pay to New York.
Since most of your tax will be paid to New York, you should continue to have your employer withhold New York tax from your pay.