They could all be correct, depending on context. For the first one it's more common to say "I don't want to hurry." You might use "hurry up" if someone specifically ordered you to hurry up and you grumpily replied, "I don't want to hurry up!"

I don't want to hurry up. (So yes it is correct as you wrote it). I don't want to be in a hurry. I was in a hurry. These are the correct ways to write those sentences, but the meaning of each is a bit different and without knowing your intentions I can't say which is the best choice. And yes, you need the article 'the' to describe the first sentence (the same way as I did just then).

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