Learn Croatian: Odabrat ću čaj umjesto kave. - I will choose tea instead of coffee.

The preposition umjesto (instead of) always requires the noun that follows it to be in the genitive case. For feminine nouns ending in -a like kava, the genitive ending is -e.

Why is the verb written as odabrat instead of the full infinitive odabrati?

In the Croatian future tense, when a verb ending in -ti comes immediately before the helping verb (like ću), it drops the final -i. If you flipped the word order or used a pronoun, it would keep the -i: Ja ću odabrati čaj.

Since čaj is the direct object being chosen, why doesn't its ending change?

It actually is in the accusative case here! However, because čaj is a masculine inanimate noun (a thing, not a living being), its accusative form is exactly the same as its nominative (dictionary) form.

Could I use the verb birati instead of odabrati here?

You use odabrati here because it is a perfective verb, indicating a single, completed action of making a choice. The imperfective form birati would sound more like an ongoing process (meaning "I will be choosing").