This paper proposes a novel relay selection scheme for buffer-aided wireless networks with relays equipped with both data buffers and energy storage. While buffer-aided relay networks have demonstrated significantly improved performance, energy harvesting has become an attractive solution in many wireless systems, garnering considerable attention when applied to buffer-aided relay networks. It is known that state-dependent selection rules must be used to achieve full diversity order in buffer-aided relay networks, requiring link priorities for data transmission to be set based on system states. This task becomes challenging when both data buffers and energy storage are involved. In this paper, we introduce a novel method for setting link priorities, which forms the basis for a new selection rule. The outage probability of the proposed selection scheme is derived. The simulation results demonstrate the superiority of our proposed algorithm which achieves full diversity in buffer-aided relay selection with energy storage, and consistently outperforms baseline approaches across various metrics.