Wouldn’t picking the dark age turn everyone with the dark sign hollow?

Going hollow is due to the dark sign as humans are meant to be immortal due to the bit of the dark sole they have but Gwin cursed them so they have a fixed life span. Flame dying means dark getting stronger which causes the humans to go back to there natural immortal state however due to the curse they can only do this by going hollow. (Or at least This is what I understand)

So surely picking the dark age would cause this to happen and everyone goes hollow?

PS - this is kinda a DS1 question but I don’t have enough karma for that sub 😬