I have (had) the same issue. I bullied my way into a solution by modifying class.misc.php around about line 66:
//$headers .= "From: ".$fromname."<".$fromaddress.">".$eol;
//$headers .= "Reply-To: ".$fromname."<".$fromaddress.">".$eol;
//$headers .= "Return-Path: ".$fromname."<".$fromaddress.">".$eol;
//$headers .= "Message-ID: <".time()."-".$fromaddress.">".$eol;
//$headers .= "X-Mailer: osTicket v 1.6".$eol;
$headers .= "From: ".$fromaddress.$eol;
$headers .= "Reply-To: ".$fromaddress.$eol;
$headers .= "Return-Path: ".$fromaddress.$eol;
$headers .= "Message-ID: ".time()."-".$fromaddress.$eol;
$headers .= "X-Mailer: osTicket v 1.6".$eol;
Getting rid of the < and > characters from $headers was the only solution I could come up with. Otherwise, some servers (not all) would show the same problem you describe.