It doesn't matter whom you report to or whom you support. Those may be organizational or budgetary considerations for the company, but they have nothing to do with income tax. As far as taxes are concerned, all that matters is what state you actually sat in when you were working, and what state you lived in.
If you got paid for the week in North Carolina, that pay is NC income and you have to pay NC tax on it. So you will have to file a NC nonresident tax return and a NJ resident tax return. You will get a credit on your NJ return for part or all of the tax that you pay to NC. Your W-2 should have two lines of state information - a line for NC showing one week's pay, and a line for NJ showing (roughly) 51 weeks' pay. You will pay NJ tax on your NJ income, which is most of your income for the year.
If your employer will not correct the W-2, you will still have the problem that the NC income shown on your W-2 is much higher than your actual NC income. If it's likely that you will end up in that situation, it probably would be a good idea to get help from a tax professional. You should go to a tax pro now, just in case there is any chance of getting corrections made before the end of the year. Also, tax professionals are not especially busy at this time of year, and going now will give the tax pro time to work on your situation.
If the employer does not retroactively correct your withholding, the NJ tax withheld on your W-2 will be much less than your actual NJ tax. That means you might have to pay a penalty for not having had enough tax withheld. The tax pro will be able to help you with strategies to possibly reduce the penalty.
And again, it's very important to get your employer to stop withholding NC tax and start withholding NJ tax going forward. A tax professional can help you with this as well. There should be no problem making that change, and you need to get it changed so that you don't have the same problem for 2023.
