Echocardiography is an essential examination for cardiac disease diagnosis, from which anatomical structures segmentation is the key to assessing various cardiac functions. However, the obscure boundaries and large shape deformations due to cardiac motion make it challenging to accurately identify the anatomical structures in echocardiography, especially for automatic segmentation. In this study, we propose a dual-branch shape-aware network (DSANet) to segment the left ventricle, left atrium, and myocardium from the echocardiography. Specifically, the elaborate dual-branch architecture integrating shape-aware modules boosts the corresponding feature representation and segmentation performance, which guides the model to explore shape priors and anatomical dependence using an anisotropic strip attention mechanism and cross-branch skip connections. Moreover, we develop a boundary-aware rectification module together with a boundary loss to regulate boundary consistency, adaptively rectifying the estimation errors nearby the ambiguous pixels. We evaluate our proposed method on the publicly available and in-house echocardiography dataset. Comparative experiments with other state-of-the-art methods demonstrate the superiority of DSANet, which suggests its potential in advancing echocardiography segmentation.