It's fine to say 'I'm in a hurry' - as in your title - but not to say 'I'm hurrying up'. You can intensify or modify this by saying 'I'm in a real hurry' or 'I'm in a bit of a hurry', or you can substitute the word 'rush' for 'hurry'. In these phrases, 'hurry' and 'rush' can be interchangeable.

In this usage "hurrying" can also be used, as in "I'm late. I'm hurrying." But no word "up"here. "Up" is used with hurry most often in the command to another, "Hurry up!" Or referring to yourself as in "I'd better hurry up."

I agree with Helen. You'd be understood. But it would be more commonly said as "I'm late, I'm in a hurry", or "need to hurry", or "gotta hurry" or any derivative thereof. Another alternative would be "I'm in a rush, we'll talk later", depending on who you talk to.

Thank you Helen and Haggen

Helen, your comment is perfect. I think you should have posted it as an answer (not a comment)!