This is the same answer I included in my solution for the "Completionist!" achievement.
The "I Need MORE Healing" achievement might take longer than you think. It unlocked for me after two complete playthroughs on Normal Mode and one playthrough on Heroic Mode. So don't worry too much, it will take some time.
Please make sure to always finish the level or your progress for this achievement won't count for the stage you just played. You know it will count towards the achievement if you get the "saved" sign showing on the loading screen after you've completed the stage.