This article proposes to utilize full-duplex (FD) communication at an unmanned aerial vehicle (UAV) device to improve the quality of service of the wireless communication systems. We analyze the considered UAV-FD systems’ performance by deriving the closed-form expressions of ergodic capacities (ECs) and energy efficiencies (EEs) in two scenarios, e.g., without and with source–destination (S–D) channel over a Nakagami- distribution. Since the joint effect of high altitude and residual self-interference (RSI) at UAV-FD significantly affects the ECs and EEs of the proposed systems, we propose an algorithm for transmit power optimization at the UAV-FD. Numerical results indicate that the EC and EE with the S–D channel are dramatically higher than the EC and EE without the S–D channel. Moreover, the EC without the S–D channel reaches a saturated ceiling when the transmit power is high enough. To deal with this issue, transmitting power optimization is an effective method. More specifically, the impacts of RSI and negative parameters are greatly reduced with the optimal transmit power. In addition, the optimal transmit power is greatly lower than the conventional transmit power. Thus, the power consumption of the considered UAV-FD systems is dramatically reduced. Importantly, the ECs and EEs with the optimal transmit power are considerably higher than the ECs and EEs without that value. Besides the RSI, the positions of UAV-FD have a strong effect on the ECs and EEs of the UAV-FD systems. Thus, the UAV-FD should fly in appropriate positions to increase the ECs and EEs. On the other hand, we thoroughly examine the impact of key parameters such as frequency, bandwidth, RSI, transmit power, transmission rate, and the horizontal and vertical positions of UAV-FD to gain valuable insights into the behavior of UAV-FD systems. Based on these insights, we propose several recommendations to enhance the ECs and EEs of UAV-FD systems. Finally, Monte-Carlo simulations are conducted to validate the accuracy of the mathematical expressions.