I'm a new mom as well and just went through the "finding a daycare" ordeal. You're right, waiting lists are long, but don't let them scare you. I was persistent in calling the centers I was most interested in, and made sure that I always spoke to the same contact (usually the director). Two of my top 3 choices became aware of openings about 2 months before I needed the care, so I actually ended up having more than one option. The the month that my baby started daycare, two other centers contacted me with last minute availability.
It is unsettling to not be able to secure a space well in advance, but don't give up, keep calling your favorite centers, and most likely something will open up.