Cross-chain asset swaps are a typical mode of asset circulation across blockchains. Existing cross-chain swap solutions, represented by the hashed timelock contract (HTLC) protocol, typically adopt a bind-then-lock mode, in which the follower is bound first and then locks assets. Under this mode, once the bound follower withdraws, the initiator cannot replace the swap counterparty and must wait for the timelock to expire before reclaiming the assets and restarting the swap process, which significantly prolongs the swap duration and incurs high on-chain costs. To address this issue, this study proposes a lock-then-bind mode for cross-chain asset swaps. In this paradigm, the follower is not pre-specified in the contract before the swap starts. Instead, an eligible responder first locks assets and is then bound as the follower. This paradigm avoids repeated retries and long-term asset locking caused by follower withdrawal. Under the new paradigm, this study proposes a cross-chain asset swap protocol based on an address signature lock, namely OpenSwap. OpenSwap designs an address signature lock by embedding the follower’s identity information into the lock structure, enabling it to be synchronized between two blockchains during the asset locking and unlocking process, thus ensuring consistency in follower binding across both chains. In addition, OpenSwap enhances protocol security and execution efficiency through the introduction of a challenge period and an assisted unlocking mechanism. Theoretical analysis and experimental results demonstrate that OpenSwap ensures atomicity while significantly reducing swap latency and lowering on-chain costs in low user response scenarios, providing a more flexible and efficient solution for cross-chain swaps.