Will my button nose change or do I need rhinoplasty?

The nose can change during puberty. Given that you already had your puberty growth spurt it is unlikely that it will correct itself. You will most likely need a correction but to be on the safe side you should give yourself another year or two to see if any change happens. RegardsDr. JDisclaimer:
This answer is not intended to give a medical opinion and does not substitute
for medical advice. The information presented in this posting is for patients’
education only. As always, I encourage you to see your personal physician for
further evaluation of your individual case.

Most of the growth of the nose is done by the teenage years.  While some tips do drop a bit over time, this is more for tip that are already heavy and low.  A tip that is too high tends not to drop significantly.  Reducing the projection and elevation of the tip would make the nose look more masculine most likely.  Best of luck to you!

If you were late to grow your nose may continue to develop until your early twenties. You still may or may not choose rhinoplasty, however given your late growth spurt there is no harm in giving things more time.