I saw this double version today when upgrading.
In my case the shortcode output was having wpautop() run on it for some reason. This means a load of extra <p> tags inserted all over the place. Not sure if this is down to a change in mejs or WP.
Try doing View Source or similar and see what your looks like. If you find some <p> tags within the mejs code then feel free to try my fix (back up first, I accept no liability, here be dragons etc…):
Edit mediaelement-js-wp.php to change (starting line 427):
$mediahtml .= <<<_end_
<{$tagName} id="wp_mep_{$mediaElementPlayerIndex}" {$src_attribute} {$type_attribute} {$width_attribute} {$height_attribute} {$poster_attribute} controls="controls" {$preload_attribute} {$autoplay_attribute} $video_skin_attribute>
{$mp4_source}
{$mp3_source}
{$webm_source}
{$flv_source}
{$wmv_source}
{$ogg_source}
{$captions_source}
<object width="{$width}" height="{$height}" type="application/x-shockwave-flash" data="{$dir}flashmediaelement.swf">
<param name="movie" value="{$dir}flashmediaelement.swf" />
<param name="flashvars" value="controls=true&file={$flash_src}" />
</object>
</{$tagName}>
<script type="text/javascript">
jQuery(document).ready(function($) {
$('#wp_mep_$mediaElementPlayerIndex').mediaelementplayer({
m:1
{$loop_option}
{$controls_option}
{$audio_size}
});
});
</script>
to
$mediahtml .= <<<_end_
<{$tagName} id="wp_mep_{$mediaElementPlayerIndex}" {$src_attribute} {$type_attribute} {$width_attribute} {$height_attribute} {$poster_attribute} controls="controls" {$preload_attribute} {$autoplay_attribute} $video_skin_attribute>
{$mp4_source}{$mp3_source}{$webm_source}{$flv_source}{$wmv_source}{$ogg_source}{$captions_source}
<object width="{$width}" height="{$height}" type="application/x-shockwave-flash" data="{$dir}flashmediaelement.swf">
<param name="movie" value="{$dir}flashmediaelement.swf" />
<param name="flashvars" value="controls=true&file={$flash_src}" />
</object>
</{$tagName}>
<script type="text/javascript">
jQuery(document).ready(function($) {
$('#wp_mep_$mediaElementPlayerIndex').mediaelementplayer({m:1{$loop_option}{$controls_option}{$audio_size}});
});
</script>
notice I removed a load of line-breaks and tabs. This solved it for me.