feucht
damp
Slightly wet or permeated by a small amount of liquid.
Wisch bitte den Tisch mit einem feuchten Tuch ab.
Please wipe the table with a damp cloth.

damp / humid
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damp
Slightly wet or permeated by a small amount of liquid.
Wisch bitte den Tisch mit einem feuchten Tuch ab.
Please wipe the table with a damp cloth.

The adjective 'feucht' (damp) belongs to the word family of 'die Feuchtigkeit' (moisture/humidity). When something is 'feucht' (damp), it contains a certain amount of moisture.
Die hohe Luftfeuchtigkeit im Bad führt zu feuchten Wänden.
The high humidity in the bathroom leads to damp walls.
In the basic meaning 'leicht nass' (slightly wet), 'feucht' (damp) is often combined with things that have absorbed water: damp hands, damp clothing, damp eyes, or a damp meadow.
Nach dem Spaziergang im Regen waren seine Socken ganz feucht.
After the walk in the rain, his socks were completely damp.
In connection with weather, climate, or air, 'feucht' (humid/damp) always means that the air has a high concentration of water. Typical combinations are 'feuchte Luft' (humid air) or 'feucht-warmes Wetter' (humid and warm weather).
An schwülen Sommertagen ist die Luft oft sehr feucht.
On muggy summer days, the air is often very humid.
In a medical context, almost exclusively in the combination 'feuchter Husten' (productive cough), the word describes a cough in which mucus is produced.
Der Arzt diagnostizierte einen feuchten Husten und verschrieb einen Hustenlöser.
The doctor diagnosed a productive cough and prescribed an expectorant.
The colloquial expression 'jemanden einen feuchten Kehricht/Dreck interessieren' means that someone is completely indifferent to something. It is very informal and expresses strong indifference.
Ob er mitkommt oder nicht, interessiert mich einen feuchten Kehricht.
Whether he comes along or not, I couldn't care less.
dry
'trocken' and 'feucht' can be opposites, but 'feucht' is not as strong as 'nass'. 'trocken' means that there is no liquid, or almost no liquid. 'feucht' means that there is a little liquid; something is slightly wet or damp. That is why a towel can be 'feucht' even though it is not really wet. The words can also be confused when talking about coughs: 'ein trockener Husten' (a dry cough) has no mucus, while 'ein feuchter Husten' (a productive cough) does. In practical terms: 'trocken' is without liquid, while 'feucht' is slightly with liquid.

Nach dem Putzen war der Boden noch feucht, deshalb warteten wir kurz.
After cleaning, the floor was still damp, so we waited a little.
Die Ärztin sagt, dass mein trockener Husten oft abends stärker wird.
The doctor says that my dry cough often gets worse in the evening.