The recent popularity of edge devices and Artificial Intelligent of Things (AIoT) has driven a new wave of contextual recommendations, such as location based Point of Interest (PoI) recommendations and computing resource-aware mobile app recommendations. In many such recommendation scenarios, contexts are drifting over time. For example, in a mobile game recommendation, contextual features like locations, battery, and storage levels of mobile devices are frequently drifting over time. However, most existing graph-based collaborative filtering methods are designed under the assumption of static features. Therefore, they would require frequent retraining and/or yield graphical models burgeoning in sizes, impeding their suitability for context-drifting recommendations. In this work, we propose a specifically tailor-made Hybrid Static and Adaptive Graph Embedding (HySAGE) network for context-drifting recommendations. Our key idea is to disentangle the relatively static user-item interaction and rapidly drifting contextual features. Specifically, our proposed HySAGE network learns a relatively static graph embedding from user-item interaction and an adaptive embedding from drifting contextual features. These embeddings are incorporated into an interest network to generate the user interest in some certain context. We adopt an interactive attention module to learn the interactions among static graph embeddings, adaptive contextual embeddings, and user interest, helping to achieve a better final representation. Extensive experiments on real-world datasets demonstrate that HySAGE significantly improves the performance of the existing state-of-the-art recommendation algorithms.