General synonyms:
* Moist
* Humid
* Wet
* Soggy
* Mildew (if referring to a specific type of dampness)
More specific synonyms:
* Slightly wet
* Slightly moist
* Dewy (for a light dampness)
* Sprinkled (if the dampness is from a light spray)
* Musty (if the dampness is accompanied by a smell)
* Foul (if the dampness is unpleasant and possibly unhealthy)
The best synonym to use depends on the specific context and the level of dampness you want to convey.