I still don't understand why there's a new moon when the moon is between the earth and the sun and a full moon when the earth is between the moon and the sun. Maybe if I stare at the model longer it'll magically make sense.
As for the full moon, the earth and sun are on the same side of the moon, so the sun is lighting the moon "from behind you" and you can see that.
In these missed eclipse the moon is usually a little too high or a little too down. The moon revolves on a plane that is tilted respect the Earth's plane of revolution. These two planes of revolution intersect at the the two points called nodes.
See this picture
https://en.wikipedia.org/wiki/File:Eclipse_vs_new_or_full_mo...
If the new moon happens to fall on the node then you will get a solar Eclipse.