Here are two pictures looking at the corner of my shower. As you can see, there is a brownish/pink mold on the grout and the drywall adjacent to the shower is pulling in moisture and becoming soft. What should be done?

https://imgur.com/a/q5kTA0K

Claude recommend using caulk, not grout, at the changes to the pan, since grout cracks, however if I do that, am I not possibly locking in water behind the tile and new caulking? Am I not better off removing the old grout and the re-applying and re-sealing?