Multi-beam low earth orbit (LEO) satellites are emerging as key components in beyond 5G and 6G to provide global coverage and high data rate. To fully unleash the potential of LEO satellite communication, resource management plays a key role. However, the uneven distribution of users, the coupling of multi-dimensional resources, complex inter-beam interference, and time-varying network topologies all impose significant challenges on effective communication resource management. In this paper, we study the joint optimization of beam direction and the allocation of spectrum, time, and power resource in a dynamic multi-beam LEO satellite network. The objective is to improve long-term user sum data rate while taking user fairness into account. Since the concerned resource management problem is mixed-integer non-convex programming, the problem is decomposed into three subproblems, namely beam direction control and time slot allocation, user subchannel assignment, and beam power allocation. Then, these subproblems are solved iteratively by leveraging matching with externalities and successive convex approximation, and the proposed algorithms are analyzed in terms of stability, convergence, and complexity. Extensive simulations are conducted, and the results demonstrate that our proposal can improve the number of served users by up to two times and the sum user data rate by up to 68%, compared to baseline schemes.