We will be moving out of New Haven in a few months but do not wish to give up our car / truck. Since I plan to do most of my shopping in CT and will be taking the train in to shop, could you give me a list of parking areas that allow long-term parking in New Haven?

There is no public long term parking in New Haven. A cars is permitted to park in a residential zone for up to 72 hours, unless otherwise posted, after that it is subject to towing. Your best option would be to pay a private lot for monthly parking.