As unmanned aerial vehicles (UAVs) are expected to play a significant role in fifth generation (5G) networks, addressing spectrum scarcity in UAV communications remains a pressing issue. In this regard, the feasibility of full-duplex non-orthogonal multiple access (FD-NOMA) UAV communications to improve spectrum utilization is investigated in this paper. Specifically, closed-form outage probability expressions are presented for FD-NOMA, half-duplex non-orthogonal multiple access (HD-NOMA), and half-duplex orthogonal multiple access (HD-OMA) schemes over Rician shadowed fading channels. Extensive analysis revealed that the bottleneck of performance in FD-NOMA is at the downlink UAVs. Also, FD-NOMA exhibits lower outage probability at the ground station (GS) and downlink UAVs than HD-NOMA and HD-OMA under low transmit power regimes. At high transmit power regimes, FD-NOMA is limited by residual SI and inter-UAV interference at the downlink UAVs and FD-GS, respectively. The impact of shadowing is also shown to affect the reliability of FD-NOMA and HD-OMA at the downlink UAVs.