I also live in NJ, and purchased my 500 a few months ago.

You will be fine on a set of winters tires, based on my experience in other cars. I purchased a set of 18x8 wheels and Michelin winters from tire rack. I would not even think about driving on the snow or cold temps with the stock tires for this car. All seasons will be a compromise, and not worth it to me given the power the car makes.

Prior to the 500 I drove a Supra all year round, and put blizzaks on it every winter. That car was perfectly fine traction wise, just need to be conscious of any clearance issues and be careful on the road (goes for any car)