Intelligent reflecting surface (IRS) assisted unmanned aerial vehicle (UAV) systems provide a new paradigm for reconfigurable and flexible wireless communications. To enable more energy efficient and spectrum efficient IRS assisted UAV wireless communications, this paper introduces a novel IRS-assisted UAV enabled spectrum sharing system with orthogonal frequency division multiplexing (OFDM). The goal is to maximize the energy efficiency (EE) of the secondary network by jointly optimizing the beamforming, subcarrier allocation, IRS phase shifts, and the UAV trajectory subject to practical transmit power and passive reflection constraints as well as UAV physical limitations. A physically grounded propulsion-energy model is adopted, with its tight upper bound used to form a tractable EE lower bound for the spectrum sharing system. To handle highly non convex, time coupled optimization problems with a mixed continuous and discrete policy space, we develop a deep reinforcement learning (DRL) approach based on the actor critic framework. Extended experiments show the significant EE improvement of the proposed DRL-based approach compared to several benchmark schemes, thus demonstrating the effectiveness and robustness of the proposed approach with mobility.