Learn Dutch: De kaas wordt schimmelig. - The cheese is becoming moldy.

In Dutch, the simple present wordt naturally translates to 'is becoming' or 'is getting'. You rarely need to use the continuous 'aan het' form with worden to express an ongoing change.

How do I change this sentence to say the cheese HAS become moldy?

You would say De kaas is schimmelig geworden. When worden is used as a linking verb meaning 'to become', it takes zijn (to be) as its helping verb in the perfect tense, and its past participle is geworden.

Is wordt being used here to make a passive sentence?

No, here worden is simply a linking verb meaning 'to become'. You can tell it is not a passive sentence because there is no past participle (like 'gegeten' or 'gemaakt') later in the sentence, just the adjective schimmelig.

Why doesn't the adjective schimmelig get an extra '-e' at the end?

Because schimmelig is used here after a linking verb (wordt). Adjectives only get an added '-e' when they are placed directly in front of the noun they describe, as in de schimmelige kaas (the moldy cheese).