Oh my gosh! That is crazy... it sounds more like my dog however, she eats everything she can fit in her mouth and even some things that don't fit so well! I would be very worried of the string being lodged in the digestive tract. I've seen wild birds that swallowed stringy materials and they didn't make it. It is a very serious issue. Do you have a local vet or wildlife rehab center you can call and ask for advice? Since it isn't rare for wild birds to have the same thing they may be a good resource for you. I used to work at the Minnesota Wildlife Rehabilitation Center and they were very helpful there... they had vets on staff. I wish you the best of luck.
