Learn French: Sa cousine est souvent occupée.

Yes! In French, possessive words like son, sa, and ses agree with the noun they are attached to, not the person who owns it. Because cousine is a feminine noun, we must use sa. Depending on the context, sa cousine can mean "his cousin" or "her cousin".

What is the word for a male cousin?

A male cousin is cousin. Because this sentence is about a female cousin, we use the feminine word cousine. This is why the sentence uses the feminine possessive sa instead of the masculine son.

Why does the word occupée end with a double "e"?

Adjectives in French must match the noun they describe. Because cousine is a feminine noun, the base adjective occupé (busy) gets an extra e tacked onto the end to make it feminine as well.

Can I put souvent before the verb?

No, in French, short adverbs like souvent (often) usually go directly after the conjugated verb. That is why the sentence says est souvent rather than souvent est.