Nope. Whether your name is Shaindy or Sunshine, Chaim or Christina, Mendy or Moustafa, you are Jewish if you’re Jewish. Now, who qualifies as Jewish is a much bigger and hotly debated question!
If you’re Jewish (by whatever definition you’re using) and don’t have a Hebrew name, you can continue to live without a Hebrew name if you want.
If you want to take on a Hebrew name, then just choose one. You don’t need any special ceremony, but you can be “officially” named during the Torah reading. If you’re a man, they’ll call you to the bimah for an aliyah and do the naming as part of the Misheberachs after your aliyah. If you’re a woman, then a man will get an aliyah for you or you can appoint the gabbai/reader/rabbi to read it during the “official” Misheberachs (after the third aliyah, if memory serves). Mi sheberachs are traditionally used to pray for the ill, but anyone who needs a blessing can be included.
The Torah service is one way of making it public. But be careful that a zealous rabbi doesn’t name you before you know it or before you’re ready. I know of one person who was named at the Torah but had no idea until the rabbi came up and excitedly said, “So did you hear me name you??” Don’t be that guy, rabbis of the world. Make sure people know beforehand. This should be special.
My only advice is to sit on your name choice for a few months (or more!) to make sure it’s “right.” If it still feels good after a few months, go for it. If you’re still uncertain, wait a little longer. Some people wait for a year or more! The only reason you may need to speed up your decision process is if you need to use that name soon. For instance, you’re getting married or having a geirus l’chumrah done (to remove any uncertainty about your Jewishness). If you think you need a Hebrew name in order to get a Jewish divorce, then I’ve got some bad news: you cannot have a marriage that requires a religious divorce without having used your Hebrew name at the marriage on a ketubah. On the other hand, that means less work and no mamzerim from your future relationships! Always look on the bright side, right?
If you need to use a “Hebrew name” in the meantime, for instance if you ask someone to pray on your behalf or need to fill out an enrollment form, you can use your English name. For example, Amanda bat Jennifer or Andrew ben Stanley. Remember, your Hebrew name is a “legal name,” so it only has to be used in very limited circumstances.
I suggest you update this post to include that when a couple gets a secular divorce, they should always have a halachic divorce where the man gives his ex-wife a get. This is true even if the wedding ceremony was conservative, reform, or secular. Rav Moshe Feinstein has an opinion that if the wedding ceremony wasn't Orthodox, there can't be mamzerus, but other Rabbis disagreed with him, and it's best to remove all doubt in such serious situations.