@Archman
In general, you pay tax to both the state in you live in and the one you work in. Your home state gives you a credit for the tax you pay to the work state. You should already know that, unless this is your first year working in NY.
Working at home (in NJ), for a NY employer is usually still considered as working in NY, for tax purposes, if you work outside New York for your own convenience, and not the convenience of the employer.*
The circumstances of this situation, are not clear. It sounds like convenience of the employer to me. But, your employer has continued to withhold NY tax. You should ask your employer for guidance. They should be in contact with the NY tax authorities. Since NY and NJ are both high tax states, it may make little or no difference.
*See : http://www.journalofaccountancy.com/issues/2009/jun/20091371.html
Here's a link to New York's memorandum on its "convenience of the employer" tax doctrine regarding non-resident telecommuters: https://www.tax.ny.gov/pdf/memos/income/m06_5i.pdf
