Background: The ongoing COVID-19 pandemic has influenced educational systems worldwide. School principals coped with numerous significant challenges regarding school management during the epidemiological crisis that could generate a lot of work-related stress. Thus, the presented study examines Polish school principals’ perceived stress and its association with exhaustion and psychosomatic complaints as burnout risk indicators. Principals’ gender and age as sociodemographic control variables were also considered in this paper. Methods: A cross-sectional online study was conducted in eight provinces of Poland from June to December 2021. The study was part of a global COVID-HL school principal survey under the global COVID-Health Literacy Research Network. Two subscales of the Perceived Stress Scale (PSS-10) (perceived helplessness [PH] and perceived self-efficacy [PSE]) were considered independent variables in relation to school principals’ mental and physical exhaustion and psychosomatic complaints. Regression models consisting of two equations were used to test the relationship between variables. The first equation consists of the control variables (age, gender), and in the second equation, the independent variables (PH and PSE) were included in addition to the control variables. Results: Almost 50% of school principals experienced a lack of control that caused anger and stress. Mental and physical exhaustion during the pandemic was often or always felt by 30% of respondents. Nearly half of Polish school principals experienced psychosomatic complaints in the form of muscle pain and headaches. PH, to a greater extent than PSE, was associated with mental and physical exhaustion and psychosomatic complaints. With age, the level of psychosomatic complaints and mental and physical exhaustion decreases, but it was higher among women. Regression analysis revealed significant associations between exhaustion and mental health outcomes, even after controlling for demographic variables Conclusion: This study showed that almost half of Polish school principals indicated a high frequency of perceived stress during the pandemic. PH was more substantially associated with mental and physical exhaustion in younger female principals than PSE. Younger female school principals reported more exhaustion and psychosomatic complaints. This finding should be the baseline information for policymakers to improve the wellbeing of Polish school principals and prevent the risk of burnout.