Heat olive oil in a large Dutch oven or heavy pot over medium-high heat, then add sausage and cook until browned, about 5 minutes, and remove to a plate.
Add chicken pieces to the same pot and cook until browned on all sides, about 6-7 minutes, then remove to the plate with sausage.
Add onion, bell pepper, and celery to the pot and cook until softened, about 5 minutes, then add garlic and cook for 1 minute more.
Stir in rice, diced tomatoes, chicken broth, Cajun seasoning, paprika, thyme, salt, and pepper, then return sausage and chicken to the pot.
Bring to a boil, then reduce heat to low, cover, and simmer for 25-30 minutes until rice is tender and liquid is absorbed.
Stir in shrimp, cover, and cook for an additional 5-7 minutes until shrimp are pink and cooked through.
Remove from heat, garnish with sliced green onions, and serve directly from the pot.