penetrating, rising

have, suffer from
The house had woodworm and rising damp.

check for, look for
find
The surveyor found damp in the kitchen.

be, feel, look, smell
The room smelled damp.

become, get
make sth
The rain had made the walls damp.

extremely, fairly, very, etc.
a little, slightly, etc.
Our clothes got a little damp.

still

Damp is used with these nouns:
air, basement, brow, cave, cellar, chill, clay, climate, cloth, curl, earth, flannel, gloom, grass, ground, hollow, patch, rag, sand, smell, soil, sponge, towel, washcloth, weather