When it's grout vs grout at a 90-degree intersection in a wet environment (wall-to-wall, wall-to-floor), you'll be the loser.
Corners in wet environments should not be grouted...the grout can't hack any flexing and will eventually, crack, allowing water to break the plane and get into the framing.
First, remove all of the caulk that you applied. Then use a hand-held grout saw or a carbide bit in a dremel-type tool to remove the grout at all of the wall intersections.
Allow the area behind the tile to dry. Use diluted bleach or a mildewcide type product to kill the existing mold and mildew.
One the area behind the walls has been cleaned up and dried, it's time to repair all of the joints. First, though, you may want to seal all of the grout lines in the shower area. It's optional, but a good time to do it.
Before caulking, use backer rod in all of the joints to provide a backstop for the caulk...then caulk.