"In New Jersey, someone who receives severance pay or a severance package from a former employer may also apply for unemployment benefits. As long as the severance is paid in a lump sum, is intended to recognize past years of service, or otherwise doesn't extend the person's employment with the company, it won't affect unemployment eligibility.
If severance pay does extend the employment relationship, however, unemployment benefits may not be available until the severance ends. For example, if you receive a lump sum amount of severance on your last day of work, you may apply for unemployment. However, if you are terminated and the company pays severance that keeps you on the company’s payroll for an additional six pay periods, you would not be entitled to unemployment compensation until your severance pay runs out.
For this reason, you may want to request that your severance be paid in a lump sum as soon as your employment ends, to make sure you will be eligible for unemployment benefits as quickly as possible."
https://www.employmentlawfirms.com/resources/employment/unemployment-and-social-security/severance-effect.htm