This issue has been solved. This hosting was on GoDaddy, not Dreamhost like I said earlier. They apparently did not have a setting enabled for my hosting that allowed multiple sessions. So when this plugin tried to create the tmp directory, it failed.
I am not clear why the previous version of the plugin didn’t have this issue, but GoDaddy has now resolved it and things are working fine. Apparently this can impact other plugins, too, so if you see a plugin with a session_start error and open …/tmp/… failed, contact your hosting provider.