Place beans in a Dutch oven or soup kettle; add water to cover by 2 in. Bring to a boil; boil for 2 minutes. Remove from the heat; cover and let stand for 1 hour. Drain and rinse beans, discarding liquid.
In a Dutch oven, cook the beef, onion and celery over medium heat until meat is no longer pink; drain. Stir in flour until blended. Gradually stir in water. Add the beans, chili powder, cumin and sugar. Bring to a boil. Reduce heat; cover and simmer for 1-1/2 hours or until beans are tender. Stir in the tomatoes, vinegar and salt; heat through, stirring occasionally.
Meanwhile, for quesadillas, spread about 1 tablespoon of chilies on half of each tortilla. Sprinkle with 1/4 cup of cheese; fold in half. In a large skillet, cook tortillas in 1 teaspoon of oil over medium heat until lightly browned on each side, adding more oil as needed. Cut each in half. Serve with chili.