How do you empty a filled soul gem? Or is it not possible? (In Skyrim you drop it and pick it back up.) I have a bunch of soul gems with lower souls than their ratings.

The only thing to do is use them but then the stone vanishes. You must get an empty one in order to fill it with the appropriate soul.

Don't worry about that too much. There's a decent supply of empty ones, and once you get Azura's Star they're just clutter in your inventory.

The only time it might be significant is if you're trying to get a Grand Soul at a low level, where there aren't many creatures that have them. It can be tricky to get from picking up a Grand gem to the location of the few creatures that will fill it. In those circumstances, the only safe option is conversion to a black soul gem and a quick visit to a bandit camp.