Zero-shot cross-domain sequential recommendation (ZCDSR) enables predictions in unseen domains without additional training or fine-tuning, addressing the limitations of traditional models in sparse data environments. Recent advancements in large language models (LLMs) have significantly enhanced ZCDSR by facilitating cross-domain knowledge transfer through rich, pretrained representations. Despite this progress, domain semantic bias -- arising from differences in vocabulary and content focus between domains -- remains a persistent challenge, leading to misaligned item embeddings and reduced generalization across domains. To address this, we propose a novel semantic bias-aware framework that enhances LLM-based ZCDSR by improving cross-domain alignment at both the item and sequential levels. At the item level, we introduce a generalization loss that aligns the embeddings of items across domains (inter-domain compactness), while preserving the unique characteristics of each item within its own domain (intra-domain diversity). This ensures that item embeddings can be transferred effectively between domains without collapsing into overly generic or uniform representations. At the sequential level, we develop a method to transfer user behavioral patterns by clustering source domain user sequences and applying attention-based aggregation during target domain inference. We dynamically adapt user embeddings to unseen domains, enabling effective zero-shot recommendations without requiring target-domain interactions...