Just looked at the latest forecast - it appears that EWR will be more of a rain/snow mixture. My experience at EWR, which is my home airport, is that international flights are given priority over domestic flights so if the forecast remains correct, you should not have a problem. My concern is that you stated you need to be "northern NJ" in the early evening. Do you know exactly where you need to be?

While the turnpike and Garden State Parkway are generally plowed immediately and are passable most of the time, when you get into the small towns, it takes longer to get the streets cleaned. Also, and this is not always, but storms hit the northern counties with more snow, it all depends on how the track moves - right now northern NJ is due for all snow and no rain.

Also, tennessetom is correct. Now that students are able to do school remotely, in NYC they call it very early to provide parents sufficient time to arrange for child care and meals. I just checked a number of school districts in northern NJ and I cannot find any that either have delayed opening or remote learning.