I don't understand your situation. Do you want to create a new protocol for your Ryu/dojo? or do you want to do a respectful/suitable behavior for Japanese guest judges? I reply to your posts as latter situation.
1.Judges,my name is Sabro.
translation: "Shinpan no sensei gata, watashi no namae wa Sabro desu".
But I think you need not call out the judges. In Japan, students in most dojos simply say "osu" before saying something. If you use "osu" in your daily keiko at your dojo, use it instead of "Shinpan no sensei gata". To talk with the judges before competing feels a bit too intimate, smells somewhat injustice for Japanese sense. However, such protocol will be different in every dojos, so don't trust my words.
2. My style is Okinawan Shuri Ryu. My Sensei is Dominic Laureano. My Kata is Wansu.
translation:"watashino ryuuha wa
Okinawa Shuri Ryu desu. Sensei wa Dominic Laureano desu. watashino Kata wa Wansu desu".
However, in Japan, we don't call it Shuri Ryu, but 首里手(Shuri-te). So I don't know the word "Shuri Ryu" is suitable or not.
3.May I begin?
translation:"hajimetemo yoroshii deshou ka?"
But in my opinion, "dewa, hajime masu(so, now I start mine)" sounds more natural for your situation.