About those C's that are flat--it makes a difference what you are doing with your throat and mouth when you play a wind instrument. Think of the instrument as an extension or amplifier for the song which you are making in your throat. Don't just leave it up to the ocarina and your fingers to make the notes come out.

It's hard to put into words, but if you listen carefully and try to modify the tone until it sounds right, it may happen. (Of course, an ocarina is pretty simple and may not be "improvable", but try it.)

Your public library probably has collections of well-known songs, I believe in the 780's. Look there for songs you like.