Flying Ad-hoc Networks (FANETs) provide essential emergency communication services in post-disaster scenarios by enabling data relay when terrestrial infrastructure is damaged. However, designing effective routing strategies presents a significant challenge, as they must simultaneously optimize data transmission efficiency and energy consumption. Poor routing decisions can create energy hotspots, leading to premature Unmanned Aerial Vehicle (UAV) failures and network fragmentation. Existing routing strategies predominantly focus on instantaneous performance metrics while neglecting long-term network survivability, often resulting in unbalanced energy consumption patterns. The core challenge lies in balancing immediate transmission efficiency with long-term energy sustainability under strict real-time constraints. This paper proposes Lyapunov-based Energy-Aware Distributed Routing (LEADR) strategy, which addresses this challenge by transforming the mission-long optimization problem into a series of low-complexity, per-timeslot decisions for each UAV. By integrating the Lyapunov drift-plus-penalty framework with geographic and energy-based heuristics, LEADR enables proactive energy balance management while maintaining high network throughput. Extensive simulations demonstrate that LEADR significantly outperforms baseline strategies, achieving superior network lifetime, aggregate throughput, energy balance, and lower packet loss rate, particularly in dense deployment scenarios.