Your payroll is set up correctly. You are a resident of Vermont who works in New Jersey. Your income is taxable to both New Jersey and Vermont. However, you are entitled to a credit on your Vermont tax return for taxes paid to another state.
Per the New Jersey Division of Taxation, all wages earned by a nonresident of New Jersey, who is employed by a business located within the state, are considered New Jersey source income for income tax purposes.
Most states assess tax on nonresident income based on the amount of time worked in that state. However, there are 5 states: Delaware, Nebraska, New Jersey, New York, and Pennsylvania where nonresident "telecommuter" income is taxed based on where the business (employer) is located. Please click on this link for more information.
Since you are a resident of Vermont, all of your income is taxable to Vermont. If you pay taxes to another state, you are entitled to credit on your Vermont return for taxes paid to another state. This prevents you from being double-taxed on the same income.
When preparing your return in TurboTax, you will need to indicate that you made money in another state (New Jersey) while you were a resident of Vermont. You will need to prepare a nonresident return for New Jersey and a resident return for Vermont.
Please complete your nonresident New Jersey return first, to ensure that your state taxes are calculated correctly. Then complete your resident Vermont return last.
Please click on the following links for more information.
TurboTax Help Article - How to file nonresident state return
Instructions for New Jersey Nonresident Return - See page 3 and page 5 (Line 15: Wages)
Vermont Department of Taxes - Credit for taxes paid to other states
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