I have a cat that eats ANYTHING that might potentially, in some way, be potentially string like. It started with tinsel. I never had a problem before with cats and tinsel, though I do know the warnings. One Christmas we started to discover tinsel hanging out of her rear end. I tried to chase her down, but she ended up hiding in the litter box and eliminated the rest of it there. So, no more tinsel for our trees. Then she progressed to thread from my sewing room. It's all in the thread box, but occasionally some little bits have been known to hang out of the drawer. So she ate them. This was discovered as she was in the process of eating it. I managed to get it away from her before she swallowed too much. She found some yarn, I am not sure where, that I discovered in a similar manner to you describe. Once I caught her trying to eat the plastic thread that holds carpets together. It's kinda like fishing line, but more flat. Anyway, she discovered a section of carpet that had been pulled up in one corner to lay cable line. That I discovered when I caught her trying to puke it up and not getting anywhere, it was hanging out of her mouth, but she had swallowed part of it so it was stuck in her throat. She couldn't swallow the rest of it, couldn't puke it up and kept gaggin on it.
That one resulted in a $100 trip to the vet, money that we didn't have and had to borrow from my teen daughter. The vet was not able to do anything really. They did just what I did first, tried to pull it out gently but couldn't get it. They did an x-ray to make sure there wasn't some sort of carpet tack or staple on the end she swallowed, there wasn't. They sedated her to see if they could pull it out that way and were not able to. So, at that point, our options were do surgery to remove it, clip it as close to the throat as they could get to make it easier to swallow the rest, then wait to see if she passes it without distress. With that we had the option of leaving her with the vet overnight for observation, that way if she became distressed they could do surgery immediately, or, take her home with a list of specific distress signs and if she showed any, take her to the emergency department so they could do surgery.
Despite the fact that I was genuinely concerned for my kitty, I had to take her home. I already had to make payment arrangements for the X-ray and that particular month was truely a matter of pay the vet or make the housepayment. And, within 24 hours, she passed it and it was fine.
Now, I just try VERY hard to keep anything sting like locked up. We are a tinsel free and easter grass free house as well. I also try to let her outside some to eat grass, hoping she will puke out there and anything she might have eaten will come up. (I don't keep cat grass inside because she will then puke inside.) so far we haven't had any other major issues, the one time some easter grass snuck in and she ate it, she managed to eliminate it quickly.