You need to add to headers:
$headers .= 'MIME-Version: 1.0' . "\r\n";
$headers .= 'Content-type: text/html; charset=utf-8' . "\r\n";
if you haven't done it yet.
Also you have to make sure your e-mail client is properly set. For example in Mozilla Thunderbird you may set it to display only TXT and then it may perform some changes to the content. Otherwise you may send email correctly but you will result you don't expect