Hi, and bienvenue to the forum! (I'm an American living in France.)

Bilingual French schools in the US are generally private schools, and chances are they are fairly limited in availability. You may want to check out the following website: New Jersey/NJ Private Schools - Private School Review where you can take a look at lists of private schools in New Jersey by county. But I suspect that the French Consulate probably has a fairly comprehensive list of those schools offering a bilingual program in the New York City area.

It may be possible to find after-school French classes for your children, or some sort of play groups in the area for French speaking children, but you won't be able to find them until you're on site.

Personally, I wouldn't decide on a place to live until you get there and have a chance to see the area. Even within the same town, there can be nice areas and not so nice areas. The real estate agents usually have lots of information about the schools in each town, both public and private (as the quality of the schools directly affects the cost of housing). Also, the school district boundaries don't always coincide with the town limits - and private schools may have school buses or other transport arrangements that cover certain areas and not others.

It's normal for the employer to pay for a few weeks of living in a hotel or "suites hotel" (i.e. with a kitchen unit) or residence hotel (basically a furnished apartment with maid service). That will give you a chance to look around the area a bit, decide where you would be comfortable living and ask questions about where other people live who work at your office.

In the NYC area, you also have to consider your public transport options (especially as gasoline prices rise!), traffic patterns, proximity to shopping, schools, health care, etc. and other considerations when deciding where to live.
Cheers,
Bev