Make the roux: Heat the flour and oil in a small saucepan or skillet over medium heat. Cook for 8 to 10 minutes, whisking constantly so it does not burn. The mixture should turn a light reddish brown. If it darkens too quickly, reduce the heat to medium-low.
Cool the roux: Remove from the heat and let it sit for a few minutes before transferring it to the bottom of a 6 to 7 quart slow cooker. Pouring it in screaming hot can crack the insert.
Layer the pot: Add the chicken, then top with the sausage, tomatoes, onion, bell pepper, celery, garlic, bay leaf, Creole seasoning, paprika, thyme, oregano, and chicken broth.
Cook: Cover and cook on LOW for 6 to 8 hours or on HIGH for 3 to 4 hours, until the chicken is cooked through and the vegetables are tender.
Add the shrimp: Turn the slow cooker to HIGH if it is not already there and stir in the shrimp. Cover and cook on HIGH for 15 to 20 more minutes, stirring twice, or until the shrimp are pink and opaque.
Finish the pot: Discard the bay leaf. Remove the chicken, chop or shred the meat, and return it to the pot. Stir in the green onions and parsley.
Season and serve: Taste and season with salt and pepper, cayenne, or hot sauce. Serve with cooked rice and garnish with additional chopped fresh parsley, if desired.