Heat a large pot over medium-high heat. Add the vegetable oil and andouille and cook, stirring, until crisp, 4 to 5 minutes. Transfer to a plate. Season the chicken with salt, then add to the pot in batches to brown, 5 minutes per side. Transfer to the plate with the andouille.
Make the roux: Remove the pot from the heat and cool slightly (there should be about 1/4 cup drippings; drain or add oil as needed). Return the pot to medium heat. Sprinkle in the flour and stir, scraping up the browned bits from the pan with a wooden spoon. Continue to cook, stirring, until the mixture smells nutty and turns a deep brown, 10 to 12 minutes. Add the celery, onion, bell pepper, garlic, cayenne, and salt to taste. Cook, stirring, until the vegetables are tender, 6 to 8 minutes.
Whisk in the broth. Return the chicken and andouille to the pot and simmer until the chicken is cooked through, about 25 minutes. Transfer the chicken to a plate and let cool slightly while the stew simmers; remove the skin and shred the meat. Return the meat to the pot. Stir in the shrimp and sherry and cook until the shrimp turn pink, 2 to 3 more minutes. Season with salt and black pepper.