Early Time Series Classification (ETSC) is critical in time-sensitive medical applications such as sepsis, yet it presents an inherent trade-off between accuracy and earliness. This trade-off arises from two core challenges: 1) models should effectively model inherently weak and noisy early-stage snippets, and 2) they should resolve the complex, dual requirement of simultaneously capturing local, subject-specific variations and overarching global temporal patterns. Existing methods struggle to overcome these underlying challenges, often forcing a severe compromise: sacrificing accuracy to achieve earliness, or vice-versa. We propose \textbf{DE3S}, a \textbf{D}ual-\textbf{E}nhanced \textbf{S}oft-\textbf{S}parse \textbf{S}equence Learning framework, which systematically solves these challenges. A dual enhancement mechanism is proposed to enhance the modeling of weak, early signals. Then, an attention-based patch module is introduced to preserve discriminative information while reducing noise and complexity. A dual-path fusion architecture is designed, using a sparse mixture of experts to model local, subject-specific variations. A multi-scale inception module is also employed to capture global dependencies. Experiments on six real-world medical datasets show the competitive performance of DE3S, particularly in early prediction windows. Ablation studies confirm the effectiveness of each component in addressing its targeted challenge. The source code is available \href{https://github.com/kuxit/DE3S}{\textbf{here}}.