On AM WCBS was up during the blackout. So was WNEW, WOR, WMCA, WEVD, WABC and most of the major AM players. I can't speak for the FM stations because I didn;t listen to any FM that night All the major AM transmitter sites have automatic backup generators. Even in the 65 blackout they were equipped with backup gens. When I was working at the WNEW transmitter site (which is now underwater in the Jersey swamps) we used to test the generator regularly. We would literally throw the main breakers to black out the station and within a few seconds you'd hear the generators starter crank and the generator start up and a few seconds later the lights, transmitter, etc. would come back up.
Yeah water was a problem. Even in buildings with water towers. When the tank is low the pumps don't operate and refill the tank.