A fun look at the meaning of "sorgen" and its prefix version and what worry has to do with getting some beer.
sorgen, die Sorge, versorgen, besorgen, besorgt
The core idea of this root was:
worrying, being sick, taking care of someone sick
The only English family member I could find in the sources is sorrow.
Sorry and sore, even though they would fit in here, come from a different root that was about suffering.