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Capturing user intent across heterogeneous behavioral domains stands as a fundamental challenge in session-based recommender systems. Yet, existing multi-domain approaches frequently fail to isolate the distinct contribution of cross-domain interactions from those arising within individual domains, limiting their ability to build rich and transferable user representations. In this work, we propose MOSAIC, a Multi-Domain Orthogonal Session Adaptive Intent Capture framework that explicitly factorizes user preferences into three orthogonal components: domain-specific, domain-common, and cross-sequence-exclusive representations. Our approach employs a triple-encoder architecture, where each encoder is dedicated to one preference type, enforced through domain masking objectives and adversarial training via a gradient reversal layer. Representational alignment and mutual independence constraints are jointly optimized to ensure clean preference separation. Additionally, a dynamic gating mechanism modulates the relative contribution of each component at every timestep, yielding a unified and temporally adaptive session-level user representation. We conduct extensive experiments on two large-scale real-world benchmarks spanning multiple domains and interaction types. The ablation study validates that each component domain-specific encoding, domain-common modeling, cross-sequence representation, and dynamic gating contributes meaningfully to the overall performance. Experimental results demonstrate that MOSAIC consistently outperforms state-of-the-art baselines in recommendation accuracy, while simultaneously providing interpretable insights into the interplay between domain-specific and cross-domain preference signals. These findings highlight the potential of orthogonal preference decomposition as a principled strategy for next-generation multi-domain recommender systems.