Robust segmenting with noisy labels is an important problem in medical imaging due to the difficulty of acquiring high-quality annotations. Despite the enormous success of recent developments, these developments still require multiple networks to construct their frameworks and focus on limited application scenarios, which leads to inflexibility in practical applications. They also do not explicitly consider the coarse boundary label problem, which results in sub-optimal results. To overcome these challenges, we propose a novel Simultaneous Edge Alignment and Memory-Assisted Learning (SEAMAL) framework for noisy-label robust segmentation. It achieves single-network robust learning, which is applicable for both 2D and 3D segmentation, in both Set-HQ-knowable and Set-HQ-agnostic scenarios. Specifically, to achieve single-model noise robustness, we design a Memory-assisted Selection and Correction module (MSC) that utilizes predictive history consistency from the Prediction Memory Bank to distinguish between reliable and non-reliable labels pixel-wisely, and that updates the reliable ones at the superpixel level. To overcome the coarse boundary label problem, which is common in practice, and to better utilize shape-relevant information at the boundary, we propose an Edge Detection Branch (EDB) that explicitly learns the boundary via an edge detection layer with only slight additional computational cost, and we improve the sharpness and precision of the boundary with a thinning loss. Extensive experiments verify that SEAMAL outperforms previous works significantly.