There is plenty to do in NY, even just in Manhatten, if you only have 6 days, I would not bother with the car as driving in the city is bonkers. There is the subway, and there are also greyhound buses and train links if you wanted to have a day out of the city. I understand with DS1 they may be more difficult, but IMO driving in central New York is not worth the likely stress!
Even some of the museums could easily take more than one day. There's a Children's Museum and a Natural History Museum (I've only been to the latter), the Met etc..
Top tip with the subways - be wary of express subs. They only stop at certain stops. It is not difficult to accidentally get on one late at night and end up on the wrong side of Harlem by accident. Easy for us to fall into as we are used to tubes and trains stopping at every stop on the line of the right colour..
Statute of Liberty and the boat etc... is worth it and good educationally. Get there early though or you will queue for an eternity. Empire State Building obviously has to be ticked off. Maybe a UN tour if your kids are that way inclined and visiting ground zero in terms of important educational sites. There's a fire museum if they're more into fire trucks etc. Visiting the original Winnie the Pooh toys at the Library.
I honestly love wandering around Central Park and there are some quieter areas where may be able to get some downtime and away from some of the noise.
There's lots of places where you can just grab a slice of pizza (think of a slice being literally like eating about a third of a standard UK supermarket pizza), or bagels, which may be less stress than a sit down meal. Many of the eateries in the touristy areas are places that will be familiar - prets, mcdonalds and TGI Fridays, or dunkin' doughnuts.
Portion sizes in eateries generally are absolutely massive, so advise going family style and sharing if you can or may end up with a simply unmanageable quantity of food, and if possible see if you can have a fridge in hotel room or use the hotel fridge as places will generally do doggy bags, which will then give you something for the next day so that you can string the meal out over more than one sitting and don't have to do the sitting down in a restaurant thing two days in a row.
Magnolia Bakery cupcake is a fun treat. The Lego Shop (which from memory is near there) is pretty epic just to go and look at it!