Dashboard Widget Form

That if('POST'...) just alone -in the middle of the plugin code- doesn’t seem right. You can hook into load-$pagenow and process the form submission there:

add_action( 'load-index.php', 'check_posted_data' );
function check_posted_data()
{
    if( 'POST' == $_SERVER['REQUEST_METHOD'] && !empty( $_POST['my_prefix_message'] ) ) {
        wp_die( "I'm here! {$_POST['my_prefix_message']}" );
    }
}

Please note that you’re missing security checks and proper prefixes to your data.

And as a side note, I would solve this with Ajax.