Learn Croatian: Želim ovaj tamni model.

You would use either, depending on what information you want! If someone asks Kakav model želiš? (What kind of model do you want?), the answer is the descriptive part: tamni model (the dark model). If they ask Koji model želiš? (Which model do you want?), they are asking you to point out a specific one, so the answer is ovaj model (this model). This sentence simply combines both pointing out and describing: ovaj tamni model.

Why is the object ovaj tamni model not in the accusative case since it is what I want?

It actually is in the accusative case! The verb želim requires an object in the accusative case. However, model is a masculine noun that is inanimate (a thing, not a living being). For inanimate masculine nouns, the accusative form looks exactly the same as the nominative (dictionary) form. Therefore, ovaj tamni model does not change its ending here, unlike a living masculine object which would change, for example: ovog tamnog psa (this dark dog).

Why is the adjective tamni instead of taman?

tamni is the definite form of the adjective, meaning 'the dark one', while taman is the indefinite form, meaning 'a dark one'. Because we are using the demonstrative pronoun ovaj (this), we are pointing to a very specific, known object. In Croatian, whenever an adjective is paired with a pointing word like ovaj, it must take the definite ending.