Keeping it covered and protected is your best bet. As far as the yellow broken skin goes, when a wound is kept covered, the skin absorbs moisture. This also happens when one stays in water for a long time as well. It is not pathological. Please continue to keep it covered withthe Neosporin until you see a doctor. Letting it air dry will not be helpful at all.
Hope this helps and good luck. Your nail will grow back, just that it will take many months to complete. It will quit hurting before long.