Graph clustering is an essential aspect of network analysis that involves grouping nodes into separate clusters. Recent developments in deep learning have resulted in advanced deep graph clustering techniques, which have proven effective in many applications. Nonetheless, these methods often encounter difficulties when dealing with the complexities of real-world graphs, particularly in the presence of noisy edges. Additionally, many denoising graph clustering strategies tend to suffer from lower performance compared to their non-denoised counterparts, training instability, and challenges in scaling to large datasets. To tackle these issues, we introduce a new framework called the Dual Adaptive Assignment Approach for Robust Graph-Based Clustering (RDSA). RDSA consists of three key components: (i) a node embedding module that effectively integrates the graph's topological features and node attributes; (ii) a structure-based soft assignment module that improves graph modularity by utilizing an affinity matrix for node assignments; and (iii) a node-based soft assignment module that identifies community landmarks and refines node assignments to enhance the model's robustness. We assess RDSA on various real-world datasets, demonstrating its superior performance relative to existing state-of-the-art methods. Our findings indicate that RDSA provides robust clustering across different graph types, excelling in clustering effectiveness and robustness, including adaptability to noise, stability, and scalability.