Online handwriting represents strokes as time-ordered trajectories, which makes handwritten content easier to transform and reuse in a wide range of applications. However, generating natural sentence-level online handwriting that faithfully reflects a writer's style remains challenging, since sentence synthesis demands context-dependent characters with stroke continuity and spacing. Prior methods treat these boundary properties as implicit outcomes of sequence modeling, which becomes unreliable at the sentence scale and under limited compositional diversity. We propose CASHG, a context-aware stylized online handwriting generator that explicitly models inter-character connectivity for style-consistent sentence-level trajectory synthesis. CASHG uses a Character Context Encoder to obtain character identity and sentence-dependent context memory and fuses them in a bigram-aware sliding-window Transformer decoder that emphasizes local predecessor--current transitions, complemented by gated context fusion for sentence-level context.Training proceeds through a three-stage curriculum from isolated glyphs to full sentences, improving robustness under sparse transition coverage. We further introduce Connectivity and Spacing Metrics (CSM), a boundary-aware evaluation suite that quantifies cursive connectivity and spacing similarity. Under benchmark-matched evaluation protocols, CASHG consistently improves CSM over comparison methods while remaining competitive in DTW-based trajectory similarity, with gains corroborated by a human evaluation.