Geo-distributed consensus protocols underpin modern databases across multiple data centers, yet their performance critically depends on effective leader management under wide-area network (WAN) dynamics. Legacy approaches fall short as they either ignore application-level factors or incur high switching overheads. This paper presents GeoLM, a lightweight, performance-oriented leader management framework. GeoLM formalizes a utility model that jointly captures network delays, node popularity, and read-write ratios, and integrates three mechanisms: (i) Target Selection to identify promising leaders, (ii) Switching Damping to bound oscillations and ensure stable leadership, and (iii) Leader Handover with log pre-synchronization to minimize downtime. GeoLM is implemented in ETCD-Raft and seamlessly extends existing protocols without altering log replication. Comprehensive evaluation-including trace-driven simulations with AWS latency traces and real-world deployments across multiple regions-demonstrates that GeoLM reduces average latency by up to 30.77%, tail latency by up to 49.75%, and improves throughput by up to 28% over baseline protocols (depending on the comparison). GeoLM further delivers consistently better performance than state-of-the-art protocols such as SwiftPaxos, CURP, and EPaxos, while maintaining strong consistency guarantees.