Hey OP.
Just wanted to say that as a non-binary person in my 30s, the comments from people claiming this is a phase that will go away/that you should "play along" are unhelpful. Non-binary people exist, they always have done, and while some people's gender identity shifts throughout life, that isn't always (or usually) the case, so please don't assume it's a phase. Take your child at their word.
For what it's worth, my life isn't only my gender identity—I have a partner, friends, kids etc. A full life. The only time my gender becomes a "big deal" or the "centre of attention" is when other people make it a big deal by refusing to use the right pronouns or deciding to argue with me about who I am. And they are invariably the ones making it an issue—it's not something I particularly love talking about.
Of course it is hard for you as a parent to have to navigate such a large shift in relation to your child, and I don't think there's anything wrong with acknowledging that to yourself/others as long, as you don't make it your child's problem (not to say you would). However, you will get there eventually, and I think asking your child to forgive any accidental slip ups in the knowledge that you are trying your hardest is fine, and I'm sure they will. If you slip up, correct yourself and move on. If it's hard at first, practice out loud to yourself or with your partner until it feels more natural (and it will before you know it).
That said, you DO have to try—always, in all situations, whether your child is present or not. If your child wishes to use they/them with all people in all situations, you must do the same. Pull people up on it (gently) when they make mistakes; take a hard line with people who refuse to accept your child.
You don't have to understand non-binary identities to actively support them—all you or anyone else needs to know is which name and pronouns to use, and whether your child would prefer terms such as "sibling" instead of more gendered ones.
You have to be your child's champion, and you can be. Don't let awkwardness or embarrassment get in the way of supporting and accepting them. It is the best gift you can give them, honestly.