General dampness:
* Moist
* Humid
* Wet
* Soggy
* Spongy
* Dewy
* Mildewy
* Musty
* Frowzy (slightly damp and stale)
Slightly damp:
* Slightly wet
* Moisten
* Drenched (but not fully soaked)
* Sprinkled
* Misty
Damp as a result of rain:
* Drizzly
* Rain-soaked
* Showered
* Dripping
Damp as a feeling:
* Chilly
* Cold (especially when referring to a damp environment)
* Uncomfortable
* Unpleasant
Figurative uses of damp:
* Dim (as in "dampened spirits")
* Subdued
* Dull
* Quiet
The best synonym for "damp" will depend on the specific context in which you are using it. Consider the nuances of each word and choose the one that most accurately reflects your meaning.