First off let me say, "Welcome to the forum!"
Elthir would be a good one to ask about that if she is on soon she might be able to point you to some links.
Looks like Julie basically means "Young" or youthful. My sources are coming up with the following.
Young is nessa youth is nessë youthful young woman is nessima
So perhaps a combination of these words or one of them. As best as I know exchanging the meaning of your name for an elvish word or name with the same meaning is the closest you can come to getting a true translation.
Personally I would do something like add "Youthful" Nessë and "Young Woman" nessima together to make something such as Nessë'Ima but I am not sure whether that would be a cannon form or not.
A popular nick name for Julie is Jewl putting Nesse and Jewel (( (= JEWEL or Silmaril) Noldomírë, Noldomír; WITH ADORNMENT OF RED JEWELS carnimírië )) together.
Since mírë is the common ending for jewel You could do something like Nessëmírë
Hope that helps or at least gives you some ideas.