As a default, I will use they/them/theirs as a gender-neutral singular pronoun. Despite what one of my English teachers tried to tell me in college, this is not incorrect grammar.
They/Them/Theirs is not the only gender-neutral singular pronoun used in the English language today, but it’s probably the most familiar to people. It’s been used for centuries (since the late 1300s). You probably use it without even realizing it in casual conversation. When you don’t know what someone’s gender is, you’ll probably say ‘them.’ In the 17th century, they tried ‘it’ but that word is dehumanizing because ‘it’ usually refers to inanimate objects and animals. It’s even more recent that grammatical texts have prescribed ‘he’ as the catch-all gender-neutral pronoun, and that creates its own problems. Because even if you mean to use ‘he’ in a generic way, people default to thinking of it as a masculine pronoun. ‘They’ is simpler and easier to use.
It is newer to use they/them/theirs to refer nonbinary individuals (since 1950), and maybe that takes some adjusting in your thinking to accept. But it’s not as difficult as learning a whole new language. When you meet someone new, you might take a little time and practice to learn their name. In the same way, it might take a little time to learn their pronouns. But if you keep reminding yourself about the right name and/or pronoun, you’ll probably pick it up before too long.
Some people may prefer other pronouns. I have seen Xe/Xem/Xyr and Zie/Hir/Hirs, and I’m sure there are more that I haven’t heard of. For my nonbinary characters, Vilqa from Divine Warriors uses They and Zil from the Fae of Calaveras series ended up using Zie/Zir/Zis. Zil may get zir own story someday to explain more. It’s a sign of respect to use someone’s preferred name and pronouns.
But really, if I’m throwing around crazy made-up fantasy words with spellings like Damondytti and Kaorinix, is it really that difficult to deal with a few extra pronouns?
You can read more about the history of singular ‘they’ on Merriam-Webster and more about gender pronouns at UW Milwaukee’s LGBT Resource Center.