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