It turns out the sleep queries were caused by session_start(). I'm using sessions to manage user data throughout the site. Our host is Network Solutions and they put the session data in a temporary directory somewhere. I've also learned that this temp directory may be accessible by anyone on that server.
I moved where my files were being saved by using session_save_path() right before calling session_start();
session_save_path("/data/#/#/#/#/#/user/#/htdocs/#/session");
session_start();
(of course, the #'s are filled in with where your values)
I have not had a sleep query and the site runs many times faster. I hope this helps anyone out there who may be having similar problems.