As a proud Jewess, it is truly commendable if you decide to adopt a Jewish one. If your personal circumstances allow, I also support your going by your new Jewish name as your official one. We are taught that one of the great merits of the Jews in Egypt was that they did not change their names to be secular, and this greatly contributed to their Exodus, which led to receiving the Torah and entering Israel.
In Jewish tradition and especially from kabbalistic sources the name of a person describes the essence of the person. That is the reason we find so much emphasis on names in classical Jewish sources. For example, God changed Sarah’s name (Gen. 17:15) from Sarai to Sarah to indicate her true, new essence as the “queen of everyone.”
While there is no obligation to have an official Jewish name, it is definitely something a Jewish person should have. Choosing a name should be easy. Here are some suggestions for choosing a name. If your name is an anglicized version of a Biblical name, choose the Biblical version. If you are named after a relative (which is an important custom in the case of righteous relative, that in a sense continues that person’s righteous “existence” in a future generation) try to find out what that person's Hebrew name was. If this relative is no longer alive, it is possible to look for the name that appears on the grave or on the ketubah marriage contract. If neither option applies, just choose a Jewish name that either sounds similar to your given name, or begins with the same letter — or any Jewish name that you like and that you feel suits you!
How to make the change? In theory you can just begin to use the new name that you have chosen and it becomes your accepted and official Jewish name. If you would like to have a more formal ceremony to commemorate the important moment you could request from the local Synagogue to recite a short prayer during the Torah-reading service in which you will be officially "named". This could be on a Monday or Thursday and doesn’t need to be specifically on Shabbat.
Whatever name you choose, enjoy it!
