Shalom and thank you for your question! You wish to choose a suitable Jewish name.
There is no limitation on which Jewish name you choose. You may choose from the entire range of Biblical names or other accepted Biblical names.
Since you were born on the festival of Shavuot, which commemorates the giving of the Torah, it was suggested to you that you might take the name Moshe, who received the Torah from G-d and taught it to the Jewish people. The festival of Shavuot also coincides with the day of King David’s passing, and it is also the birthday of the holy Baal Shem Tov, (Master of the Good Name,) founder of the Chassidic movement, who emphasized the importance of serving G-d with joy. His name was Yisrael.
Your own name is actually a Biblical name, and the Lubavitcher Rebbe often recommended to people who wanted to choose a Jewish name, that it be similar to the name they grew up with, for example:
Jack could become Jacob, Rose could become Shoshana (which means rose,) Gary could become Gavriel… and so on.
There is significance to the fact that a certain name was chosen for you. It is Divine Providence. So, perhaps you might want to simply add a name to your existing one, and be Michael Moshe for example.
It’s fine if you prefer not to.
Perhaps it can be mentioned here that if possible, one should remain on good terms with one’s biological parents, according them respect and appreciation for the gift of your life and whatever other investment of love and support they gave you.
If you do have a relationship with them and they see that you don’t reject the Biblical name they gave you, they will feel validated.
Again, it is entirely your choice. Once you have chosen, the full name should be used for occasions such as being called up to the Torah, and marriage. Moreover, from the moment you are called to the Torah with your new Jewish name, the
Hebrew letters that make up your name are like a spiritual formula of the energies that make up your soul.
When G-d created the world (Chassidic teaching is that G-d constantly recreates the world also, for if not it would cease to exist, ) it was according to the twenty two letters of the Hebrew alphabet. They are the spiritual formula of the energies that make up the world.
May you lead a life in which you are able to use those special energies that are unique to you, to the fullest!
