The feeling of wetness comes from the way water adheres to your skin. Adhesion happens when the water molecules interact with the molecules on the surface of your skin. Any substance that does this can give a feeling of "wetness."
Try this experiment to see it in action!
Get something that will let you put water on something one drop at a time. First, put a drop of water on a piece of plastic or a nonstick pan. You should see the drop bead up on the surface, because there is low adhesion. Then, do the same on a paper towel, and it should soak in due to high adhesion. Finally, put a drop on your skin and you should get something in between.
Next, let's try (safely) modifying your skin surface by putting some cooking oil on a patch of your skin. See what happens when you put a drop of water on oiled skin vs. not. Also, notice that it actually feels different too!
And now, here's a brain teaser!
If you were a fish, would you feel wet in the water? Add your thoughts in the comments!