Here is my question. When moving off of an asteroid does it count as moving through the asteroid?

If the movement template crosses the asteroid token, then yes. Otherwise no.

You do exactly what the rules say. If the movement template or the ship's final position overlap an asteroid, then it counts as an overlap and you act accordingly.

I always thought this a little strange. If you move and run over two roids you only roll for damage once. But if you land on one and your template over laps it on your next move you roll for damage a second time for the same rock.

That's because if your template hits it next time you've run into again or really 'harder' than you would have wanted to as you didn't get by it the first time.