A good planner never likes it when things go wrong, but usually triedss build into the plan some contingencies. I too book my flights well in advance when I use ff miles. Normally that is 9 months in advance (or more if busness class to Australia). I always call 2 months in advance to update the flights. Almost without exception times are changed, flight numbers are changed...and some time equipment changes as well as connetion issues. I too do not like it, but that is the way it is.
I then call a week in advance, and usually the day before.
I can only imagine how much trouble this change is for you, traveling with children! I agree CO is doing you well to supply hotel vouchers for 3 rooms! I to would try for food vouchers too...good luck. I would skip asking for a flight on another airline, it would seem unreasonable and may cost you something that they are offering. Try to documment, as possible, the room offers and any others to make receiving them straight foreward.
Someone suggested visiting Mahattan. I have not been there but we will be in a few weeks and plan to go to the Central Park Wildlife Center. Prices are $3.50 for adults and 0.50 for children. They are open 10am -5 or 5:30. There is also the Bronx zoo. It is much more expensive, but is free on Wednesdays according to the info I got on the web. I do not know how much time you will have.
Enjoy your trip, and try to lighten up on the airline. As suggested elsewhere they would have eventually told you. If they tell you too soon, you run the risk of having to change your plans another time. They also could have offered you no vouchers, only an "I am sorry for any inconvenience this may cause you".