Since their announcement in November 2020, COVID-19 vaccines were largely debated by the press and social media. With most studies focusing on COVID-19 disinformation in social media, little attention has been paid to how mainstream news outlets framed COVID-19 narratives compared to alternative sources. To fill this gap, we use cognitive network science and natural language processing to reconstruct time-evolving semantic and emotional frames of 5745 Italian news, that were massively re-shared on Facebook and Twitter, about COVID-19 vaccines. We found consistently high levels of trust/anticipation and less disgust in the way mainstream sources framed the general idea of "vaccine/vaccino". These emotions were crucially missing in the ways alternative sources framed COVID-19 vaccines. More differences were found within specific instances of vaccines. Alternative news included titles framing the AstraZeneca vaccine with strong levels of sadness, absent in mainstream titles. Mainstream news initially framed "Pfizer" along more negative associations with side effects than "AstraZeneca". With the temporary suspension of the latter, on March 15th 2021, we identified a semantic/emotional shift: Even mainstream article titles framed "AstraZeneca" as semantically richer in negative associations with side effects, while "Pfizer" underwent a positive shift in valence, mostly related to its higher efficacy. "Thrombosis" entered the frame of vaccines together with fearful conceptual associations, while "death" underwent an emotional shift, steering towards fear in alternative titles and losing its hopeful connotation in mainstream titles. Our findings expose crucial aspects of the emotional narratives around COVID-19 vaccines adopted by the press, highlighting the need to understand how alternative and mainstream media report vaccination news.