Consensus is critical for distributed databases as it ensures the consistency of states across nodes, reinforcing the robustness of the overall system. However, faults related to the consensus protocols such as Paxos can lead to serious issues in distributed databases. Such consensus issues impact the correctness and availability of these databases. Therefore, to automatically uncover consensus issues in distributed databases, we propose Conan, a framework designed with fuzzing-driven fault injection. Conan applies a state-guided fuzzing algorithm to effectively explore the fault search space. Moreover, Conan employs hybrid fault sequences that combines fine-grained message-level faults and coarse-grained system-level faults to enhance fault injection. We implement and evaluate Conan on 3 widely-used distributed databases, including etcd, rqlite and openGauss. Finally, Conan has successfully uncovered previously unknown consensus issues, some of which are not detected by existing approaches.