In this paper, we study a full-duplex (FD) unmanned aerial vehicle (UAV) communication system affected by residual transceiver hardware impairments (RTHI) and residual self-interference (RSI). We derive analytical expressions for the symbol error probability (SEP) and achievable rate (AR) under the separate and combined effects of RTHI and RSI. To mitigate the negative impact of these impairments, particularly RSI inherent in FD operation, we propose an optimal UAV transmit power scheme that minimizes the SEP or, equivalently, maximizes the AR, subject to RTHI, RSI, and a practical transmit power constraint. The derived expressions are validated through Monte-Carlo simulations. Numerical results show that both RTHI and RSI significantly affect system performance. When one impairment is small, the effect of the other becomes more noticeable. When both are strong, SEP and AR rapidly saturate. In such cases, the proposed power control strategy helps reduce SEP, increase AR, and prevent saturation at high transmit power levels. The results also show that using optimal transmit power improves the performance of FD systems compared to conventional half-duplex (HD) systems. Importantly, the optimal transmit power is much lower than the traditional power commonly used by UAVs. As a result, the full-duplex unmanned aerial vehicle (FD-UAV) system with RTHI consumes less power than the scenario without power optimization. We also analyze the impact of modulation order, bandwidth, UAV altitude, and operating frequency on SEP and AR. Based on the results, we provide useful recommendations to improve the performance of FD-UAV systems under RTHI and RSI.