I had a cat that ate things he shouldn't and I'm sorry, but it sounds exactly like your cat's symptoms. We had to do a $1,500 surgery to remove the obstruction. And then he did the same thing 6 months later and we had to put him to sleep. He suffered from PICA which he ate anything he could find...toys, shoe laces, anything that could fit in his mouth.
You need to take him to the vet and yes, tell the vets he likes to snack on string.