Can a broken nose cause a droopy tip or bulbous tip?

I would really need to examine you to determine what your options would be.  Of course at your age you need parental consent.  I have performed rhinoplasty in girls as young as 15, usually a few years older for boys.

Rhinoplasty can be a tool to address these issues if you have completed growth.  I
would search for the board certified plastic surgeon who you feel is
most likely to give you the best results and who performs hundreds of
these procedures each year. I would look at patient reviews as well as
before and after photos on that surgeon's website.Kenneth Hughes, MDLos Angeles, CA

A rhinoplasty will correct your nasal fracture, remove the hump, refine and raise your nasal tip. See a very experienced rhinoplasty surgeon who does natural looking noses for best results.  

Thank you for your question but without photos and/or an examination it is possible to answer.  But fixing a bump alone will not lift your tipDr Corbin

Dear brownieswithicecream, Fracturing your nose may cause some abnormalities in the appearance of your nose. However, the bulbous tip is probably just a hereditary factor you were born with and it is coming more noticeable as you age. Rhinoplasty may address these concerns and you will need to speak with your parents to seek consultation in these matters since you are a minor. Upon your parents consent you may schedule consultations with surgeons to fully answer your questions and concerns. Best regards, Michael V. Elam, M.D.

Nasal Trauma may cause an abnormal position of parts of our nose. This would not explain your bulbous tip. There are many things that can be done to improve your nose; I will make suggestions after you submit pictures.

Please share this answer with your parents if you are under 18 years of age.  Breaking the nose can shift not only the nasal bones but the cartilage as well.  Pushing or flattening the nasal cartilages could further weakening their inherent tensile strength contributing to the wide and droopy look you describe.  Rhinoplasty to refine the nasal shape would address these issues.  Hope this helps.