How to prevent dry chicken in soup?

Don't cook the chicken pieces for so long. Add them ten or twenty minutes before serving. For that matter, I wouldn't simmer the aromatics for that long either.

Do the long simmer and cooking to make the chicken stock, then strain the now-tasteless and mushy expended bits from the flavorful stock. Heat the stock and add the sweated veggies, diced chicken, and any pasta/rice/dumplings desired but only cook briefly before serving.

MgGee in his Keys To Good Cooking, recommends a couple of things that go straight to the problem you describe.

First, he recommends working with lower temperatures (his bolds, his italics).

Searing meat does not seal in its juices, and moist cooking methods do not make meats moist. Juiciness depends almost entirely on how hot you cook the center of the meat. If it gets much hotter than 150°F/65°C, it will be dry.

Secondly, he recommends brining as a means of ensuring lean poultry retains moisture.

Brining is the immersion of meat in a weak solution of salt and water, with or without other flavorings, for hours to days before cooking. Injecting brine into the meat interior speeds the process. The salt penetrates the meat, seasons it, and improves its ability to retain moisture and tenderness.

Brines of a certain strength, 5 to 10 percent salt by weight, also cause the meat proteins to absorb extra water from the brine, making the meat seem exceptionally juicy when cooked. Very lean poultry and pork can benefit from this extra moisture, especially when they’re overcooked.

Don't BOIL, only ever a slow gentle simmer - Not just Chicken all Meat.

If you insist on putting your Chicken in there at the start of cooking you make be better off with boneless thighs as they contain slightly more fat and sinew (good for keeping meat moist).

Personally though I'd recommend buying a whole chicken. Shop's around me charge around the same for a whole Chicken as they do for 2 breasts. Buying a whole chicken means you'll have two (if not more) options to consider.

1. Make a roast dinner with it, then use the left over's for your soup over the next days. Just pick the carcass clean of all it's good meat, and use the bones and 'nastier' bits for your stock. Then just stir the already cooked meat into your soup right at the end.

2. Use the carcass for your stock, thighs for the soup, wings for the freezer (Kept till BBQ season marinaded in nice sticky BBQ sauce.. I digress) and the breasts for a different meal or for your soup near the end if you want lots of meat.

I have always put my aromatics and a organic free range skin-on hen in an oven safe stock pot, covered with cold (!) water, heated it on medium until it is just ready to boil, then popped it in a 300degree oven for 4-5 hours. I then strain all the veg and chicken (with the broth going into a new soup pot). Once the strained chicken is cool, I take off the Cooked chicken meat, discard the aromatics, add The shredded chicken meat to the stock, season (e.g. more salt or pepper) and then refrigerate overnight. In the morning, some (crazy) people strain the fat off the top but you’ll be missing out on the best flavour, imho. I dice up fresh carrot, celery, shallots and 2 packages of chicken breasts. I bring the soup to a gentle boil and I add the new diced chicken and veg not long before serving so the chicken just cooks through and the veg are al dente. Good luck!