How do I stop every post from going out for a while?

Press Pause posting on the Posts page. Nothing scheduled for that brand goes out, and nothing can be posted now, until someone presses Resume posting. With every brand shown, it pauses them all.

Posts keep their times while posting is paused. When you resume, a post whose time passed in the meantime goes back to drafts, so nothing goes out late on its own. Add it to the queue again when you're ready.