Domain adaptation aims to transfer knowledge from a labeled source domain to an unlabeled target domain with different distributions. In real-world scenarios, the label spaces of the two domains often have an inclusion relationship, where some classes exist only in one domain but not the other. These non-overlapping classes are referred to as private classes. Identifying private class samples and mitigating their adverse effects is critical in the literature. Existing methods rely on the assumption that shifts in private classes are large enough to be considered outliers. However, the variance within a single shared class can be significantly larger than the difference between a private class and another shared class, challenging this assumption. Consequently, private classes substantially increase the difficulty of cross-domain classification. To address these issues, based on local transportation and metric properties of optimal transport (OT), a locality-aware private class identification approach is proposed in the form of a score function on transport mass. The effectiveness of the proposed approach is theoretically proven, highlighting the score function's strong ability to distinguish between shared and private class samples. Building on this, we introduce a reliable OT-based method (ReOT) for domain adaptation under severe label shift. ReOT minimizes classification risk while learning the separated cluster structure between the identified shared classes and private classes, effectively avoiding mismatch between shared-private sample pairs, thus ensuring that important knowledge is reliably transported intra-class to mitigate class-conditional discrepancy. Furthermore, a generalization upper bound of the target risk is provided for extreme label shift scenarios, which can be minimized by ReOT. Extensive experiments on benchmarks validate the effectiveness of ReOT.