The answer depends on your own desires and feelings, on your external circumstances, and on what eventually happens.
If believe in Judaism but do not feel a longing to be study the entire Torah (Written and Oral and, more broadly, the various commentaries since), be bound by and in practice actually fulfill all the mitzvot that apply to you, and to become a part of the Jewish people, then you should remain a Noahide.
But if you long to do more, to study and fulfill the Torah and fully join the Jewish people, then that is a sign or indication that you should do so, with the eventual objective of converting according to Jewish law.
Only once your (halachically-valid, that is, Orthodox) conversion is complete will you know if you had a Jewish soul all along. If you do accomplish a valid conversion, then certainly it was meant to be and you had a Jewish soul all along. (This assumes that this idea of converts being born with Jewish souls is correct, and assumes a more mystical view of the world -- it's possible a more rationalist-leaning Orthodox Jew wouldn't see it this way.)
If due to some reason (like your location or family relationships) you are unable to convert for the time being, or even even, then it is possible that is an indication you don't have a Jewish soul or are not meant to convert.
But if you still have the desire to move closer to conversion, then don't worry about whether you have a Jewish soul, just try to adjust your life circumstances to make conversion possible, and keep trying for the conversion, refusing to take no for an answer. Sometimes it may take years.
In any case, be patient and have faith that Hashem, who does everything for the best, put you in the situation you're in for a good reason, that this is where you need to be right now for your soul to accomplish what it is supposed to accomplish in this world. It may be that you can serve God best as a Jew, and that that is what you are meant to do, but you can also serve God right now, through prayer and good deeds.
By the way, it doesn't matter whether you have any Jewish ancestry. Most converts don't. If you continue the Noahide path for now, there are many organizations and resources available. As you probably already know, there is a Noahide prayerbook (siddur) now, called Service From the Heart. There are also an increasing number of Jewish books meant for all people, like Rabbi Shalom Arush's Universal Garden of Emuna.