Payment channel networks (PCNs) enhance blockchain scalability by enabling off-chain payments, where Hashed Time-Locked Contracts (HTLCs) ensure the security of multi-hop payments across intermediaries. As the scale of payment channel networks expands, the number of transactions naturally increases, while concurrent executions on overlapping channels trigger resource contention for the limited capacity of shared channels, potentially leading to deadlocks. Based on the principle that consistent execution ordering across overlapping channels prevents deadlocks, we systematically solve the deadlock problem by formalizing deadlock formation conditions, designing detection methods based on these conditions, followed by developing total-order and partial-order sorting strategies to break potential cyclic dependencies, and ultimately proposing a fairness-aware, deadlock-free scheduling mechanism that enhances transaction success rates. Extensive simulations validate our approach, demonstrating competitive transaction success rates alongside robust deadlock prevention.