Is water wet?

The problem is how our body sense sensations. "Wetness" isnt a property of chemistry unless you start talking about the combination of hydrogen and oxygen together.

So technically you are right. However our experience of how water molecules behave gives rise to the experience of "wetness".

If you had a puddle of mercury in your hand it would take a bit longer to soak into your skin but you dont feel "wetness". On a side note dont test this without knowing mecury can soak into your skin and it is toxic.

Wetness is a quality given to something by a liquid, not a property of the liquid itself.

Do you mean "does water feel wet"?

The water is "wet" with skin not without it.
On the other hand water indeed has sticking properties. That of course is not enough to say its "wet".
By the way, you were targeting the wrong crowd by asking a scientist. Only a Philosopher is actually qualified to answer it...