Well, I got some ideas from a friend and finished the edge with a binding. I just feel like quilts should be bound. I don't think I'd like a facing or anything like that.
But my friends suggestions were to treat the side edge like it was a zig-zag instead of curves. I turned the outie points like I normally do corners and the innie points were done by clipping and spreading to make a straight line. I think it turned out really nice. Here's a picture of how the edges turned out.
Thanks for all the ideas though. I always can count on you all to help me out when I need it.