Collaborative Filtering (CF) is a foundational approach in recommender systems, but it struggles with challenges such as data sparsity and the cold-start problem. Cross-Domain Recommendation (CDR) has emerged as a promising solution by leveraging dense domains to improve recommendations in sparse target domains. However, existing CDR methods face significant limitations, including their reliance on overlapping users as a bridge between domains and their inability to address domain sensitivity, i.e., differences in user behaviors and characteristics across domains, effectively. To overcome these limitations, we propose CGSP, a unified and adaptive CDR framework based on graph signal processing (GSP). CGSP supports both intra-domain and inter-domain recommendations while adaptively controlling the influence of the source domain through a simple hyperparameter. The framework constructs a cross-domain similarity graph by integrating target-only and source-bridged similarity graphs to capture both intra-domain and inter-domain relationships. This graph is then processed through graph filtering techniques to propagate and enhance local signals. Finally, personalized graph signals are constructed, tailored separately for users in the source and target domains, enabling CGSP to function as a unified framework for CDR scenarios. Extensive evaluation shows that CGSP outperforms state-of-the-art baselines across diverse cross-domain settings, with notable gains in low-overlap scenarios, underscoring its practicality for real-world applications.