I asked Amanda, the owner of crazymomquilts blog, if she'd had similar problems with the
ticker tape quilt she made 4 years ago. This was her response to me:
"I didn't free motion quilt my ticker tape pieces on, I used a walking foot to attach the pieces. I didn't use fusible, I just sewed the pieces on, which quilted the piece at the same time. The edges are supposed to fray. If they get stringy, just trim them up! It's not supposed to be perfect.
I hope that helps!"