Significant inter-individual variability limits the generalization of EEG-based emotion recognition under cross-domain settings. We address two core challenges in multi-source adaptation: (1) dynamically modeling distributional heterogeneity across sources and quantifying their relevance to a target to reduce negative transfer; and (2) achieving fine-grained semantic consistency to strengthen class discrimination. We propose a distribution-aware multi-source domain adaptation network (DAMSDAN). DAMSDAN integrates prototype-based constraints with adversarial learning to drive the encoder toward discriminative, domain-invariant emotion representations. A domain-aware source weighting strategy based on maximum mean discrepancy (MMD) dynamically estimates inter-domain shifts and reweights source contributions. In addition, a prototype-guided conditional alignment module with dual pseudo-label interaction enhances pseudo-label reliability and enables category-level, fine-grained alignment, mitigating noise propagation and semantic drift. Experiments on SEED and SEED-IV show average accuracies of 94.86\% and 79.78\% for cross-subject, and 95.12\% and 83.15\% for cross-session protocols. On the large-scale FACED dataset, DAMSDAN achieves 82.88\% (cross-subject). Extensive ablations and interpretability analyses corroborate the effectiveness of the proposed framework for cross-domain EEG-based emotion recognition.