Data-driven storytelling has gained prominence in journalism and other data reporting fields. However, the process of creating these stories remains challenging, often requiring the integration of effective visualizations with compelling narratives to form a cohesive, interactive presentation. To help streamline this process, we present an integrated authoring framework and system, DataWeaver, that supports both visualization-to-text and text-to-visualization composition. DataWeaver enables users to create data narratives anchored to data facts derived from "call-out" interactions, i.e., user-initiated highlights of visualization elements that prompt relevant narrative content. In addition to this "vis-to-text" composition, DataWeaver also supports a "text-initiated" approach, generating relevant interactive visualizations from existing narratives. Key findings from an evaluation with 13 participants highlighted the utility and usability of DataWeaver and the effectiveness of its integrated authoring framework. The evaluation also revealed opportunities to enhance the framework by refining filtering mechanisms and visualization recommendations and better support authoring creativity by introducing advanced customization options.