Priceline only guarantees a room for two people with one bed. This is clearly not going to work for you.
While I understand you want to get accommodation settled, finalising your choice in January may not give you the best offer possible.
The UN coming back to Sept every year signals the start of the incremental rise of hotel rates for the rest of the calendar year.
In your shoes, I would research hotels to come up with a short list. I would then sign up to each hotel's website for newsletters and offers. In addition, I would sign up for Travelzoo updates as well as looking periodically on Quikbook.
If you choose to book in January, make sure it's a cancellable rate so you can take advantage of future offers.
I know The Beacon Hotel on the UWS is popular with families.
Thanks for that, initially I wanted to book accommodation as early as possible but I understand that might yield the best deal.
I do have suscription to both Quickbook and Travelzoo, but you are right, I should get myself on mailing lists of few hotels as well.
Cheers