> but will using JWT to keep the user logged in instead of sessions/cookies

You can't use JWT without cookies, or local storage (which is effectively the same here), so that makes no sense.

> Also, I do not plan on opening the API to other sites, so OAuth is unnecessary. Is my understanding correct?

You're still so vague that it's hard to answer. If you want to authenticate them, you can use OAuth. But no, it's not necessary to authenticate them with any one auth service...