My question is simply if I can earn the black star AFTER I earned azura's star.

I realized the black star is much better, but unfortunately it was after I randomly chose to bring the star back to azura's shrine instead of Neclar.

Is there any way I can re-do the quest, or earning the black star while having Azura's?

Yes, by loading a previous save . But the more you level up the more black soul gems you'll find. I wouldn't change my Azura Star against the Black Star..