At the advice of several individuals, I've done an online search for hostels and shelters in my area. I've found a hostel in a vaguely close location, but all the shelters I've found are about fifty miles away. And some of those are as bad as my parents - they probably wouldn't hit me for talking back to them, but being religious institutions there'd be a lot of pressure on me to convert.
Still, it's not all bad. I've found an interfaith place (their allies include a Unitarian Universalist church), a day support place, and apparently I can get $2 showers at the YMCA. This would basically require running all over Boise, but I think it'll be doable. (If it comes to that. As I said, this entire thing is a last-resort plan ONLY. There are other, more reasonable methods that I would like to follow instead.)