This article introduces a new approach to address the spectrum scarcity challenge in 6G networks by implementing the enhanced licensed shared access (ELSA) framework. Our proposed auction mechanism aims to ensure fairness in spectrum allocation to mobile network operators (MNOs) through a novel weighted auction called the fair Vickery-Clarke-Groves (FVCG) mechanism. Through comparison with traditional methods, the study demonstrates that the proposed auction method improves fairness significantly. We suggest using spectrum sensing and integrating UAV-based networks to enhance efficiency of the LSA system. This research employs two methods to solve the problem. We first propose a novel greedy algorithm, named market share-based weighted greedy algorithm (MSWGA) to achieve better fairness compared to the traditional auction methods and as the second approach, we exploit deep reinforcement learning (DRL) algorithms, to optimize the auction policy and demonstrate its superiority over other methods. Simulation results show that the deep deterministic policy gradient (DDPG) method performs superior to soft actor critic (SAC), MSWGA, and greedy methods. Moreover, a significant improvement is observed in fairness index compared to the traditional greedy auction methods. This improvement is as high as about 27% and 35% when deploying the MSWGA and DDPG methods, respectively.