Sensitive Information Exposure (SIEx) vulnerabilities (CWE-200) remain a persistent and under-addressed threat across software systems, often leading to serious security breaches. Existing detection tools rarely target the diverse subcategories of CWE-200 or provide context-aware analysis of code-level data flows. Aims: This paper aims to present SIExVulTS, a novel vulnerability detection system that integrates transformer-based models with static analysis to identify and verify sensitive information exposure in Java applications. Method: SIExVulTS employs a three-stage architecture: (1) an Attack Surface Detection Engine that uses sentence embeddings to identify sensitive variables, strings, comments, and sinks; (2) an Exposure Analysis Engine that instantiates CodeQL queries aligned with the CWE-200 hierarchy; and (3) a Flow Verification Engine that leverages GraphCodeBERT to semantically validate source-to-sink flows. We evaluate SIExVulTS using three curated datasets, including real-world CVEs, a benchmark set of synthetic CWE-200 examples, and labeled flows from 31 open-source projects. Results: The Attack Surface Detection Engine achieved an average F1 score greater than 93\%, the Exposure Analysis Engine achieved an F1 score of 85.71\%, and the Flow Verification Engine increased precision from 22.61\% to 87.23\%. Moreover, SIExVulTS successfully uncovered six previously unknown CVEs in major Apache projects. Conclusions: The results demonstrate that SIExVulTS is effective and practical for improving software security against sensitive data exposure, addressing limitations of existing tools in detecting and verifying CWE-200 vulnerabilities.