So, something kind of weird happened. It looks like it's all fixed, but I'm not exactly sure how; here's what I did.
I went in and added some dummy published posts, in the hopes of forcing Wordpress to think there was a second page. Sure enough, that worked - my second page loaded fine. Except that caused some problems of its own - the second page had a button that went to the third page, and the third page, which should have had a post on it, didn't load. So, not perfect, but maybe workable.
Then I went in and changed the number of posts per page to 10 in the query. I intended to use 10 in the live site, so I figured I'd just troubleshoot the proper number rather than messing around with it too much.
And it works now. The second page loads, and doesn't have a "next" button on it. I don't know what fixed it, exactly, but it appears to be fixed. Thanks very much for both your help.