Site migrated to the new server with PHP 7.2 and I can't log in
We found the problem. The site was using HTTPS on the production server and HTTP on the test server, and we had set php_flag session.cookie_secure On in our Apache config. HTTPS wasn't in use, so no cookies were sent, so no logins would work properly. Setting this to Off worked wonders. Thanks for pointing me in the right direction, folks!