The COVID-19 pandemic catalyzed an unprecedented surge of misinformation on social media, frequently intertwined with emotionally charged language. Understanding both the sentiment and truthfulness of this content is critical for public health monitoring and misinformation mitigation. However, Bangla—despite being a globally prominent language—remains severely underrepresented in joint sentiment and fake news detection research, with existing studies largely restricted to single-task settings. To bridge this gap, this paper proposes a novel multi-task BanglaBERT-based framework for the simultaneous classification of sentiment and truthfulness in COVID-19 discourse. Furthermore, we introduce the first publicly available, dual-annotated Bangla corpus for this domain, comprising 35,526 textual samples aggregated from social media and news sources. Our architecture employs a shared BanglaBERT encoder with dual task-specific heads, optimized using a task-prioritized loss function that combines modified Focal Loss and weighted cross-entropy to address inherent class imbalances. Extensive experiments demonstrate that the proposed model achieves 75.1% accuracy (Macro F1: 0.707) for sentiment classification and 88.0% accuracy (Macro F1: 0.851) for truthfulness detection. Ablation studies and error analyses confirm that our tailored loss strategies significantly enhance the recognition of underrepresented and semantically ambiguous classes, particularly neutral sentiments. By releasing our dataset, code, trained models, and a Gradio-based interactive demo, this work establishes a robust benchmark for multi-task learning in low-resource Bangla NLP and provides a practical tool for fact-checking during health crises.