We develop a joint offloading and resource allocation framework for a multi-layer cooperative fog computing network, aiming to minimize the total energy consumption of multiple mobile devices subject to their service delay requirements. The resulting optimization involves both binary (offloading decisions) and real variables (resource allocations), making it an NP-hard and computationally intractable problem. To tackle it, we first propose an improved branch-and-bound algorithm (IBBA) that is implemented in a centralized manner. However, due to the large size of the cooperative fog computing network, the computational complexity of the proposed IBBA is relatively high. To speed up the optimal solution searching as well as to enable its distributed implementation, we then leverage the unique structure of the underlying problem and the parallel processing at fog nodes. To that end, we propose a distributed framework, namely feasibility finding Benders decomposition (FFBD), that decomposes the original problem into a master problem for the offloading decision and subproblems for resource allocation. The master problem (MP) is then equipped with powerful cutting-planes to exploit the fact of resource limitation at fog nodes. The subproblems (SP) for resource allocation can find their closed-form solutions using our fast solution detection method. These (simpler) subproblems can then be solved in parallel at fog nodes. The numerical results show that the FFBD always returns the optimal solution of the problem with significantly less computation time (e.g., compared with the centralized IBBA approach). The FFBD with the fast solution detection method, namely FFBD-F, can reduce up to $60\%$ and $90\%$ of computation time, respectively, compared with those of the conventional FFBD, namely FFBD-S, and IBBA.