We propose a novel approach to assess the public's coping behavior during the COVID-19 outbreak by examining the emotions. Specifically, we explore (1) changes in the public's emotions with the COVID-19 crisis progression and (2) the impacts of the public's emotions on their information-seeking, information-sharing behaviors, and compliance with stay-at-home policies. We base the study on the appraisal tendency framework, detect the public's emotions by fine-tuning a pre-trained RoBERTa model, and cross-analyze third-party behavioral data. We demonstrate the feasibility and reliability of our proposed approach in providing a large-scale examination of the publi's emotions and coping behaviors in a real-world crisis: COVID-19. The approach complements prior crisis communication research, mainly based on self-reported, small-scale experiments and survey data. Our results show that anger and fear are more prominent than other emotions experienced by the public at the pandemic's outbreak stage. Results also show that the extent of low certainty and passive emotions (e.g., sadness, fear) was related to increased information-seeking and information-sharing behaviors. Additionally, high-certainty (e.g., anger) and low-certainty (e.g., sadness, fear) emotions during the outbreak correlated to the public's compliance with stay-at-home orders.