Learn Dutch: Zij drukt haar verdriet uit. - She expresses her sorrow.

The verb being used here is the separable verb uitdrukken (to express). In a standard main clause, the conjugated part (drukt) takes the second position, while the separable prefix (uit) is pushed to the very end of the clause.

Is there a reason why verdriet doesn't use the -de suffix to become a noun, like liefde or koude?

Yes! The -de suffix you are currently learning about is used to turn adjectives into nouns (like the adjective lief becoming liefde). Verdriet, however, is not derived from an adjective. It comes from the verb verdrieten (to sadden or grieve), so it doesn't take that suffix.

Could I also use the verb uiten here?

Yes, Zij uit haar verdriet is also perfectly natural. Uiten and uitdrukken are very close in meaning when talking about emotions. However, uitdrukken is a bit broader; you can also use it for expressing thoughts, concepts, or expressing yourself in a language (zich uitdrukken).

Why doesn't haar get an -e ending here, even though we often add -e to modifiers before a noun?