How can I add a class to particular paragraphs in my post?

I figured this out. Even though WordPress automatically adds the <p> tag and doesn’t show it to you in the editor (in either the ‘visual’ or ‘text’ editor views), you can type it in anyway in the ‘text’ editor view, along with a class.

So to do this, go to the ‘text’ editor view, and just type, e.g., <p class="section-break>Here is the first paragraph of a new section of my document</p>.

And although when you type a regular <p> tag yourself (without a class) it will disappear when you switch to the ‘visual’ editor and back to the ‘text’ editor view, if instead you type it in with a class, it will stay there available to edit if it has a class attribute (e.g., <p class="something">stuff</p>).