Societies worldwide have witnessed growing rifts separating advocates and opponents of vaccinations and other COVID-19 countermeasures. With the rollout of vaccination campaigns, German-speaking regions exhibited much lower vaccination uptake than other European regions. While Austria, Germany, and Switzerland (the DACH region) caught up over time, it remains unclear which factors contributed to these changes. Scrutinizing public discourses can help shed light on the intricacies of vaccine hesitancy and inform policy-makers tasked with making far-reaching decisions: policies need to effectively curb the spread of the virus while respecting fundamental civic liberties and minimizing undesired consequences. This study draws on Twitter data to analyze the topics prevalent in the public discourse. It further maps the topics to different phases of the pandemic and policy changes to identify potential drivers of change in public attention. We use a hybrid pipeline to detect and analyze vaccination-related tweets using topic modeling, sentiment analysis, and a minimum of social scientific domain knowledge to analyze the discourse about vaccinations in the light of the COVID-19 pandemic in the DACH region. We show that skepticism regarding the severity of the COVID-19 virus and towards efficacy and safety of vaccines were among the prevalent topics in the discourse on Twitter but that the most attention was given to debating the theme of freedom and civic liberties. Especially during later phases of the pandemic, when implemented policies restricted the freedom of unvaccinated citizens, increased vaccination uptake could be observed. At the same time, increasingly negative and polarized sentiments emerge in the discourse. This suggests that these policies might have effectively attenuated vaccination hesitancy but were not successfully dispersing citizens' doubts and concerns.