How to get out of an unjust library fine?

At a public library, I ordered a lot of books online to check out. After I had finally read all of them, I dropped them off in a Book Drop. The book drop is the big canister outside the library that you place books in when you can't go inside. A couple weeks after wards I began receiving bills for books that I KNOW I had already turned in. When I called the library to see what had happened, they said that they had never received my books and that they can't look for them because they were from different libraries over the state. After discovering this, I became livid. I felt so betrayed that the library I liked so much could be so irresponsible as to forget to check my books in before they sent them back. Of course the library won't hardly ever confess to this horrid act, I know that it is the most likely explanation as to what happened. I don't steal and I've been disgusted with it. The only proof I have is my mother. I'm seventeen years old and she drove me to the book drop to drop them in. My family doesn't have a lot of money so paying the fine will be quite difficult. I highly doubt my mother would hire a lawyer for legal advice. Is there anything I can do about this other than paying that unjust fine? I really would do everything I could to pay the fine if I had truly lost or stole the books. However, I refuse to try hard to pay anything for something that I shouldn't have to pay. My mom says I should try to at least pay something to the fine because it could turn into me getting arrested and gaining a record. PLEASE HELP ME!!!!