Error rate analyses of cooperative non-orthogonal multiple access (CNOMA) systems are of paramount importance to investigate the communication reliability for each user and facilitate the development of enhancement algorithms. Although CNOMA has recently attracted great attention, its error performance, particularly that of full-duplex cooperative NOMA (FD-CNOMA), is still underexplored in the literature. In this paper, we investigated the error performance of FD-CNOMA systems under imperfect successive interference cancellation (SIC) and residual self-interference (RSI), where new closed-form expressions of the exact bit error rates (BER) are derived for both users. Through the derived BER expressions, high-SNR analyses are conducted to show that FD-CNOMA has an error floor. Based on the derived expressions, we proposed a novel SINR-based selective FD-relaying, which minimizes the end-to-end (e2e) BER and improves the overall system performance. The analyses are extended to cover pulse-amplitude modulation (PAM) and quadrature-amplitude modulation (QAM) with arbitrary modulation orders. Monte Carlo simulations and numerical results are presented to corroborate the derived analytical expressions and give valuable insights into the error performance of FD-CNOMA systems.