Bring 3 cups chicken broth and rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until the rice is tender and liquid has been absorbed, 20 to 25 minutes.
Heat cast-iron Dutch oven over medium-high heat. Cook and stir andouille sausage in hot pot until browned, about 5 minutes. Remove sausage from the pot with a slotted spoon, retaining any drippings in the pot.
Saute sweet onion, green onions, celery, and bell pepper in the sausage drippings until tender, 5 to 7 minutes. Season vegetable mixture with Creole seasoning, minced garlic, hot sauce, and black pepper; cook for 1 minute more.
Pour tomato sauce and diced tomatoes over the vegetable mixture; stir and add shrimp, shredded chicken, browned sausage, and 1 cup chicken broth into the tomato mixture. Bring the mixture to a boil, reduce heat to medium-low, and cook at a simmer until he shrimp are no longer translucent, 10 to 15 minutes.
Scoop rice into bowls and ladle jambalaya over the rice.