Try spraying with a light mist of water to soften the stabilizer. Then tear off the project. There is also wash away stabilizers that might be easier to take off.
Are you using the right needle/thread combo in the machine? What type of thread are you using and how is it threaded onto the machine? Certain threads need some distance with unwinding. A thread stand behind the machine lets the thread unwind without kinks. Are you pulling, tugging on the project or stitching too fast?
Any of the above can contribute to thread breakage. It's a wonder we ever get to stitch...LOL!