Learn German: Meine Socken sind nass.

In German, adjectives only take an ending when they come directly before the noun they describe (like meine nassen Socken). When the adjective comes after a verb like sein (to be) to describe the subject, it acts as a predicate adjective and stays in its basic, uninflected form.

What is the singular form of Socken?

The standard singular is die Socke (feminine). Interestingly, in parts of southern Germany, Austria, and Switzerland, you might hear people use a masculine singular form instead: der Socken.

Could I use feucht instead of nass here?

You could, but it changes the meaning. Nass means truly wet or soaked (like you stepped in a deep puddle). Feucht means damp or moist (like a towel that has been hanging up to dry for a few hours). Here, the socks are thoroughly wet.