Coflow has emerged as a fundamental application-layer abstraction in distributed systems, representing communication dependencies and enabling collaborative management of related flows to enhance job completion efficiency. To meet the increasing bandwidth demands of modern data center networks (DCNs), optical circuit switches are widely deployed due to their high capacity and energy efficiency. Simultaneously, DCN deployments are evolving towards heterogeneous parallel architectures, where multiple independent optical circuit switching (OCS) cores operate concurrently to facilitate bandwidth expansion and incremental upgrades. However, existing research on coflow scheduling in multi-core switching fabrics primarily focuses on electrical packet switching (EPS) networks, with a few known results on OCS networks without or with a poor performance guarantee. This paper studies the coflow scheduling problem in multi-core OCS networks under the not-all-stop (i.e., asynchronous) reconfiguration model, focusing on two major challenges of overcoming cross-core coupling for inter-core traffic allocation and satisfying the constraints of port exclusivity and reconfiguration overhead for intra-core circuit scheduling. To minimize total weighted coflow completion time (CCT), we propose an efficient algorithm by integrating linear programming-guided (LP-guided) global coflow ordering, inter-core flow allocation and intra-core circuit scheduling that achieves approximation ratios of 8K and 8K+1 for zero and arbitrary release times of coflows, respectively, where K is the number of OCS cores. This framework is also applicable to H-core EPS networks, providing approximation guarantees of 4H and 4H+1 for zero-time and arbitrary-time release, respectively.