Have you sent them the book already? If not, then don't send it to them; keep making excuses and see if you can get out of your contract that way. If you've sent them the file for the book then they're going to publish it.
I'm sure you're creative, and I'm sure another story will come to you soon. I've been working on one of my books for a long while, and it is a lot of work, especially revising it, but it's probably not something I'll send to publishers or agents because I dont' think it's good enough. That's a learning experience. You write to become much better, and us creative types will always have another story brewing in the backs of our heads. So, start working on a new and better book, because there's nothing you can do right now; PA's contract is legally binding. Always research any publisher and agent you're thinking about sending your work, too, because many pretend to be legit agents or real publishers, and sometimes they're not. PA pretends to be a "traditional" publisher, but they're a vanity press. Major difference between the two terms.
Oh, they're right, your book will be available through Barns and Noble, Borders, too, and many other bookstores. BUT, that's part of their deceptive tactics. What they don't tell you is that it will be available through their websites ONLY. Many people see that and think their books will be stocked on their shelves. It won't. In fact, you'll find that most bookstores will not stock PA books because PA has a bad reputation in publishing. Most bookstores will not stock POD (print on demand) books, and PA is a POD. Sorry this happened. The only thing you can do is start writing a better story and try to get it published with a real publisher. Good luck.
http://www.aaronline.org has a list of legit literary agents. Most large publishers will only accept manuscripts from a literary agent. One more thing, what kind of children's book did you write? There are different kinds, and for one type of children's book there isn't a market for it because people aren't buying them, so publishers arent' really publishing them. For another type of children's book it might be better to get it published without an agent. Is it Young Adult, Middle Grade, picture book, chapter book, first reader, or what? They are all classified as "children's books" and they are all different types of books.
Here's a forum thread where a literary agent answers questions about children's books, and she gives insights into the market for them. You might find this info helpful. http://absolutewrite.com/forums/showthread.php?t=115508 There are also other threads like it in that section (ask the agent) and you can glean more about the publishing industry by reading those threads. You also might want to seek out literary agent and publisher blogs. THey have a lot of info about what publishers are looking for and why manuscripts get rejected. I suggest you start following those blogs so that you can increase your chances of being accepted. Good luck.
http://nathanbransford.blogspot.com - agent blog
http://pubrants.blogspot.com - agent blog
http://behlerblog.wordpress.com - publisher blog
There are many more like them, and they usually link to each other, so you shouldn't have a problem finding some you'd like to follow.