General wetness:
* Moist - this is a more formal and neutral term
* Damp - slightly wet, not soaked
* Soggy - very wet and heavy, like after a downpour
* Muddy - wet and covered in mud
Specific situations:
* Waterlogged - saturated with water, can refer to soil or land
* Dewy - covered in morning dew
* Rain-soaked - drenched by rain
* Flooded - covered with water, usually to a significant depth
The best word will depend on the specific situation and the level of wetness you want to convey.