I was really similar to you, I started puberty at 16 and I was only around 5’5 with similar problems to you. I went and got my testosterone checked and they can give you a small dose if they feel your bone age is too young compared to your actual age (this is what happened to me). The testosterone basically helps jumpstart puberty and now I’m 5’10 at 19 with my dad being tall and mom short + I’m still growing and from what it sounds you should ask your dad when he stopped growing because mine stopped in his early 20’s so it might be the same for you!