Heat oil in 6-quart Dutch oven over medium-high. Cook andouille and chicken 5 minutes, stirring a few times to brown some surfaces.
Cook onion, bell pepper, and celery 5 minutes. Add garlic, paprika, thyme, cayenne, and salt 30 seconds.
Measure tomato juices in jug; add only enough broth for 2 3/4 cups total liquid. Stir rice and drained tomatoes into pot; add measured liquid.
Simmer; cover tightly and cook over low 18 minutes without stirring.
Nestle shrimp into surface; cover and cook 5 to 7 minutes until shrimp opaque and 145°F, chicken 165°F, and rice tender.
Remove from heat; rest covered 5 minutes. Gently fold to distribute shrimp; serve in 6 bowls.