In this paper, a novel reconfigurable intelligent surface (RIS)-aided full-duplex (FD) cooperative non-orthogonal multiple access (CNOMA) network is investigated over Nakagami-m fading channels, where two RISs are employed to help the communication of paired users. To evaluate the potential benefits of our proposed scheme, we first derive the closed-form expressions of the outage probability. Then, we derive users' diversity orders according to the asymptotic approximation at high signal-to-noise-ratio (SNR). Simulation results validate our analysis and reveal that users' diversity orders are affected by their channel fading parameters, the self-interference of FD, and the number of RIS elements.