Plugin is creating posts twice

The code does not adds posts twice – checked in my local installation.

You have got errors in it:

$my_post = array(
    'post_title'    => $title,
    'post_date'     => $insDate,
    'post_content'  => '$url',
    'post_status'   => 'public',
    'post_type'     => 'post',
);

Should be

$my_post = array(
    'post_title'    => $title,
    'post_date'     => $insDate,
    'post_content'  => $url,
    'post_status'   => 'publish',
    'post_type'     => 'post',
);

Otherwise, the problem will be in the form posting the variables. Maybe it get’s redirected and is submited twice, or something similar.