How should I check what type of texture of hair I have? When wet or dry?

When you let it dry naturally. Water can weigh down your hair texture, when it's dry and if it's wavy / curly it can bounce up a bit. 

I would say when dry as well, as you can see the curl pattern/texture more easily, because water can weigh down curls. I would suggest applying a defining product before its dry to really define any curl or wave.

Dry.  When its wet, all you feel is wet hair, but when it is dry, you can see how thick or thin it is.

maybe you can go to a hair salon and have them check it