Ultrasound Strain Elastography (USE) is a powerful non-invasive imaging technique for assessing tissue mechanical properties, offering crucial diagnostic value across diverse clinical applications. However, its clinical application remains limited by tissue decorrelation noise, scarcity of ground truth, and inconsistent strain estimation under different deformation conditions. Overcoming these barriers, we propose MUSSE-Net, a residual-aware, multi-stage unsupervised sequential deep learning framework designed for robust and consistent strain estimation. At its backbone lies our proposed USSE-Net, an end-to-end multi-stream encoder-decoder architecture that parallelly processes pre- and post-deformation RF sequences to estimate displacement fields and axial strains. The novel architecture incorporates Context-Aware Complementary Feature Fusion (CACFF)-based encoder with Tri-Cross Attention (TCA) bottleneck with a Cross-Attentive Fusion (CAF)-based sequential decoder. To ensure temporal coherence and strain stability across varying deformation levels, this architecture leverages a tailored consistency loss. Finally, with the MUSSE-Net framework, a secondary residual refinement stage further enhances accuracy and suppresses noise. Extensive validation on simulation, in vivo, and private clinical datasets from Bangladesh University of Engineering and Technology (BUET) medical center, demonstrates MUSSE-Net's outperformed existing unsupervised approaches. On MUSSE-Net achieves state-of-the-art performance with a target SNR of 24.54, background SNR of 132.76, CNR of 59.81, and elastographic SNR of 9.73 on simulation data. In particular, on the BUET dataset, MUSSE-Net produces strain maps with enhanced lesion-to-background contrast and significant noise suppression yielding clinically interpretable strain patterns.