I implemented the recommended changes, restarted the webserver, cleared all browsing data, and tried different browsers with the same results. Adding the section you recommended I modify just to make sure that looks as you intended.
function __construct($name, $ttl = SESSION_TTL, $checkdbversion = false) {
// session name/ssid
if ($name && strcmp($this->name, $name))
$this->name = $name;
// Session ttl cannot exceed php.ini maxlifetime setting
$maxlife = ini_get('session.gc_maxlifetime');
$this->ttl = min($ttl ?: ($maxlife ?: SESSION_TTL), $maxlife);
// Set osTicket specific session name/sessid
session_name($this->name);
// Set Default cookie Params before we start the session
session_set_cookie_params($this->ttl, ROOT_PATH, Http::domain(),
osTicket::is_https(), true);