Federated Learning (FL) is a promising distributed machine learning framework that allows collaborative learning of a global model across decentralized devices without uploading their local data. However, in real-world FL scenarios, the conventional synchronous FL mechanism suffers from inefficient training caused by slow-speed devices, commonly known as stragglers, especially in heterogeneous communication environments. Though asynchronous FL effectively tackles the efficiency challenge, it induces substantial system overheads and model degradation. Striking for a balance, semi-asynchronous FL has gained increasing attention, while still suffering from the open challenge of stale models, where newly arrived updates are calculated based on outdated weights that easily hurt the convergence of the global model. In this paper, we present {\em SEAFL}, a novel FL framework designed to mitigate both the straggler and the stale model challenges in semi-asynchronous FL. {\em SEAFL} dynamically assigns weights to uploaded models during aggregation based on their staleness and importance to the current global model. We theoretically analyze the convergence rate of {\em SEAFL} and further enhance the training efficiency with an extended variant that allows partial training on slower devices, enabling them to contribute to global aggregation while reducing excessive waiting times. We evaluate the effectiveness of {\em SEAFL} through extensive experiments on three benchmark datasets. The experimental results demonstrate that {\em SEAFL} outperforms its closest counterpart by up to $\sim$22\% in terms of the wall-clock training time required to achieve target accuracy.