Place chicken in a Dutch oven; add water to cover. Bring to a boil; cover and simmer 1 hour or until tender. Remove chicken from broth; cool. Reserve 1 cup broth. Bone chicken, and coarsely chop.
Cook spaghetti according to package directions; drain and set aside.
Sauté onion, celery, and garlic in oil in a large Dutch oven until tender. Add flour; cook until bubbly. Stir in tomatoes and reserved broth; bring to a boil. Add Worcestershire sauce, seasonings, shrimp, mushrooms, and reserved chicken; mix well. Gently fold spaghetti into chicken mixture.
Spoon half of spaghetti-chicken mixture into a greased, shallow 2 1/2-quart casserole. Sprinkle with 1/2 cup Parmesan cheese. Top with remaining spaghetti mixture. Dot with butter; cover and bake at 350° for 30 minutes. Sprinkle with remaining Parmesan cheese; bake, uncovered, 15 minutes.