Hello,
Thanks for attaching the file. The problem is that you are trying to apply a character style to a paragraph tag. Although addText allows it, the embedHTML can't apply it. You need to use a paragraph style, for example using a custom paragraph style (DocHeading4) from your document:
$docx->importStyles('word.docx', 'merge', array('DocHeading4'));
$wordStyles = array('.dblunderline' => 'DocHeading4');
$docx->embedHTML('<p class="dblunderline">my test</p>', array('strictWordStyles' => true, 'wordStyles' => $wordStyles));
$docx->createDocx('result.docx');
Please note that you can't apply a character style using embedHTML and a Basic license. You'd need to use HTML Extended available only in Premium licenses.
Regards.