If you have a reasonable manager, then yes, I would disclose to them and ask for time off to cope and seek treatment. Your manager can tell your team that you had an emergency health issue to deal with. Which is true! In American work culture, health issues are generally understood to be private, so people are unlikely to pester you about it.
If you don't trust your manager to keep your diagnosis private, then you can try telling them that you have a pressing health issue that you don't feel comfortable discussing in detail. That might be enough for them to give you time off. Or you can just say that it's sudden-onset burnout.
Depending on your company, looping in HR might help. If it's a big organization or has been around for a while, they've likely dealt with similar circumstances before.
Best of luck, and I hope you feel better soon.