If you’re collecting $ from NJ you cannot work, it’s illegal. Unless you’re getting cash obviously

"The N.J. Family Leave Act forbids employees on leave from performing services full-time for any employer for whom the employee did not provide those services before beginning FLA leave. An employee on FLA leave may work part-time, but not for more than half as many hours weekly as the employee regularly works for the leave-granting employer. Employers may not forbid such part-time work."

So if you want to work for your current employer, you can, but for not more than half as many hours as you worked before. I had an employee who did this while on NJ Family Leave. You can't work at all for any other employer either full-time or part-time. It's pretty straightforward. You can wrestle with the cash options on your own.

https://nj.gov/labor/myleavebenefits/help/faq/fli.shtml

NJ FLI link above. Note that FLI and FMLA/NJFLA are different programs but often run concurrently. FLI is wage replacement, the others provide for protected leave for qualified employees. I'm not an expert but I did use FLI for bonding time in 2020 when the program was a 6 week max with a weekly benefit max of $881. It's now 12 weeks and 85% of salary up to $1025/wk for 2023 (must make $1206/wk+ to maximize).

All that being said, I don't think there's any advantage to working part-time while on FLI unless you job isn't protected, and it's some sort of deal with your employer to keep your job. The state FAQ doesn't address it.