The chicken in my chicken soup turns into little stringy pieces--i.e. it doesn't stay in nice bite-sized pieces. It happens in chicken/rice and chicken/noodle recipes. Is there a way of avoiding this?
Yes, don't cook the chicken the entire time in the soup. Any chicken that is simmered long enough to create broth or stock is going to be very well cooked, and will shred easily.
Instead, near the end of the cooking, add fresh diced chicken, that poaches in the broth. You can also cook it separately, and add it at the time of service.
It sounds like you are badly overcooking the chicken. Try first cooking the browned chicken pieces in the broth for a very short time, just until done, then getting them out and cooking the rest of the soup in the broth. Add the chicken pieces back to the pot after you have turned the stove off.
The above paragraph assumes that you are cutting white meat (chicken breast) into neat bites. I can't imagine this to work for dark meat (chicken thighs), but if you are using dark meat, you should cook it for a long time and live with the fact that it looks ugly. White meat gets tough and dry when overcooked, besides falling apart into strings; dark chicken meat is tough if it is just cooked, it only gets pleasantly soft after it has reached the stage where it shreds by itself.