You have two different situations, so we will address each separately.
How do I file my state tax returns? It is really unfortunate that the company withheld PA tax. PA and NJ have reciprocity, so even if this is PA income, you don't pay PA tax on it. However, in order to get PA tax refunded to you, you will need to file a PA nonresident return with $0 income. TurboTax will help you to do this in the PA state return.
Since NJ is your home (resident) state, all of your income will be taxable there.
The wild card is NY. If you have previously filed NY nonresident returns for work performed in NY, and you physically worked in NY this year, you will continue to pay tax on physical work performed in NY. What is absolutely unknown is what NY is going to argue about the telecommuting income. NY is the most aggressive state in considering telework for a NY-based (and located) company as NY income. Because of your specific parameters, I hesitate to guess and certainly do not assume what NY's position will be. Here is why I think it's a bit complicated:
As you state, your company was paying you as an independent contractor and then changed to paying you as an employee. Obviously, they did not register your pay as taxable in NY. They could be right, but it's no guarantee that they are. And if the telecommuting income in question was performed on behalf of the NY-based location, NY likely (again, not going to assume here) will consider this income to also be taxable to NY state as well.
You may need to investigate a little further with the company to make certain. They should be able to tell you if the work you performed remotely is attributable to NY or not. If it is, then the safe route is to report it as nonresident income to New York and pay the tax on it. New Jersey will give you a credit for the tax you pay to New York instead of to New Jersey.
How do I file if they previously paid me as a contractor and switched to a W-2 employee? This also happens with some companies. However, if you are treated as an employee for part of the year, you can file a form (available in TurboTax), that requests to treat you as an employee the entire year. Here is a link with more information:
Form 8919 - Internal Revenue Service
If you do receive Form 1099-NEC (new this year instead of Form 1099-MISC) for your contract work, you may choose to use Form 8919 so that the "independent contractor" portion of the income is treated as wages. You would be responsible for 1/2 of the Social Security/Medicare due on that income, and your company the other half. (If under $600, you might not receive the Form, but you still should claim the income. You might choose not to worry about Form 8919, however, if the amount of money involved is that little).
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