damp:  The familiar adjectival use of damp as ‘slightly wet’ is a comparatively recent development, from the 18th century. When the word was first borrowed into English, from Middle Low German damp, it was a noun meaning ‘vapour’ (an application which survives in fire-damp). It comes ultimately from a Germanic base *thump-. The first line of semantic development taken by the word in English was of a ‘noxious exhalation’ (including gas or even smoke, not just vapour), and this is reflected in its earliest adjectival use, in the late 16th century, meaning ‘dazed’, as if affected by such harmful fumes; ‘with looks downcast and damp’, John Milton, Paradise Lost 1667.
Another contemporary sense was ‘noxious’. But the 17th century saw the noun used more and more for specifically wet turbidity: ‘mist’, or simply ‘moisture’. And this formed the basis of the present-day adjectival sense.
early 14c., "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 (cognates: Old High German damph, German Dampf "vapor;" Old Norse dampi "dust"). Sense of "moisture, humidity" is first certainly attested 1706.