Fallacies are arguments that seem valid but contain logical flaws. During the COVID-19 pandemic, they played a role in spreading misinformation, causing confusion and eroding public trust in health measures. Therefore, there is a critical need for automated tools to identify fallacies in media, which can help mitigate harmful narratives in future health crises. We present two key contributions to address this task. First, we introduce FALCON, a multi-label, graph-based dataset containing COVID-19-related tweets. This dataset includes expert annotations for six fallacy types-loaded language, appeal to fear, appeal to ridicule, hasty generalization, ad hominem, and false dilemma-and allows for the detection of multiple fallacies in a single tweet. The dataset's graph structure enables analysis of the relationships between fallacies and their progression in conversations. Second, we evaluate the performance of language models on this dataset and propose a dual-transformer architecture that integrates engineered features. Beyond model ranking, we conduct statistical analyses to assess the impact of individual features on model performance.