I disagree with the way ability score increases. I'd honestly do Constitution +2 and +1 to one other stat (evidence suggests that Vampires vary a lot, some have inhuman strength, speed, grace, or some even say intellect)
The fangs ability and the recovery is very cool, however the mechanic you're using doesn't really exist in 5e which prefers to use other mechanics.
I would say it would be better if it dealt say 1d8 necrotic damage and you recovered half, increasing to 2d8 at 5th level, 3d8 at 11th and 4d8 at 17th.
I would keep the restriction to once per short or long rest the same.
I could possibly see other traits for this race, but this is fairly simple and straightforward.