To whet is to sharpen. To wet is — well, you know what it is. To moisten, soak, etc. So not really applicable to your appetite.
Something could dampen your appetite but that means to suppress or decrease, not to increase. In that context, dampen is not a synonym for moisten but related to the noun damper, as a restraint or device that suppresses something — such as the flow of air to a fire, or the sound coming from a musical instrument.
And then there’s “wet your whistle” in which “wet” means “wet” (but not dampen, as in the sense seen above). Phrases.org traces this saying back to the 14th century.
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