I think we all need a refresher from biology classes about the central dogma of biology. This is the principle that the flow of genetic information goes from DNA -> mRNA -> proteins. This video does a great job illustrating this important principle (it's short, you should watch it).

https://www.youtube.com/watch?v=9kOGOY7vthk

The mRNA Covid vaccines inject mRNA strands (represented by the yellow strands in the video), which contain the blueprints for building proteins. The mRNA contained in these vaccines is a small part of the SARS-CoV-2 genome and codes for just the outer spike proteins of the virus, which let's the virus recognize and bind to cell tissue. Your cells take the mRNA from the vaccine and builds spike proteins, which your immune system then learns to recognize and attack as foreign. The vaccine DOESN'T introduce DNA. It DOESN'T splice mRNA into the DNA. In fact, it doesn't change or affect your DNA at all. The vaccine doesn't even introduce any parts of the virus, alive or dead. Only a sequence of mRNA code for a very specific portion from the virus' genome.