Mobile edge caching (MEC) has received much attention as a promising technique to overcome the stringent latency and data hungry requirements in the future generation wireless networks. Meanwhile, full-duplex (FD) transmission can potentially double the spectral efficiency by allowing a node to receive and transmit in the same time/frequency block simultaneously. In this paper, we investigate the delivery time performance of full-duplex enabled MEC (FD-MEC) systems, in which the users are served by distributed edge nodes (ENs), which operate in FD mode and are equipped with a limited storage memory. Firstly, we analyse the FD-MEC with different levels of cooperation among the ENs and take into account a realistic model of self-interference cancellation. Secondly, we propose a framework to minimize the system delivery time of FD-MEC under both linear and optimal precoding designs}. Thirdly, to deal with the non-convexity of the formulated problems, two iterative optimization algorithms are proposed based on the inner approximation method, whose convergence is analytically guaranteed. Finally, the effectiveness of the proposed designs are demonstrated via extensive numerical results. {It is shown that the cooperative scheme mitigates inter-user and self interference significantly better than the distributed scheme at an expense of inter-EN cooperation. In addition, we show that minimum mean square error (MMSE)-based precoding design achieves the best performance-complexity trade-off, compared with the zero-forcing and optimal designs.