First off welcome to the QB from SE Michigan. I am sorry you are having this difficulty with your first quilting project. The blocks nor any piecing should be coming apart. I would ask about it. I just hope this doesn't sour you to quilting.hopefully you will find the passion for making your own. It is not uncommon, as mentioned by others to use pieces of the receiving blankets for the binding. This sounds like a larger quilt. Was it quilted by a professional on a larger quilting machine or by a friend or someone using a domestic machine as a kindness to get your project completed with love? If this was done by a professional that person should stand behind their work. It may or may not be something you want to complain about. You are justified however.
I do hope to see you here. It would be fun to help you to begin learning and loving this joy of quilting. Again welcome to the QB!