early 14c., "a noxious vapor," perhaps in Old English but there is no record of it. If not, probably from Middle Low German damp; ultimately in either case from Proto-Germanic *dampaz (cf. Old High German damph, German Dampf "vapor;" Old Norse dampi "dust"). Sense of "moisture, humidity" is first certainly attested 1706.
late 14c., "to suffocate," from damp (n.). Figurative meaning "to deaden (the spirits, etc.)" attested by 1540s. Meaning "to moisten" is recorded from 1670s. Related: Damped; damping.
1580s, "dazed," from damp (n.). Meaning "slightly wet" is from 1706. Related: Dampness.