Be sure to inform your payroll department that you are now an NJ resident.
Your NY employer is not obligated to withhold NJ taxes if you meet all three of the following:
• Employed totally outside New Jersey;
• Subject to the withholding tax of that state; and
• The withholdings required by that state equals or exceeds the withholdings required for New Jersey.
The income you earn following your move is taxable by both states, but you'll be able to take a credit on your NJ return for the taxes you pay to NY. The credit can be taken only up to the amount of NJ tax you'd owe on that income if you had earned it in NJ. (In other words, if your credit exceeds the amount of your NJ taxes due, NJ won't refund you the excess amount.)
Since NY's tax rates tend to be higher than NJ's, if the job is your only source of income you should be OK with only NY taxes being withheld.