The do_shortcode function accepts a string, searches the string for shortcodes, and processes any that are found. That means that…
// Use shortcode in a PHP file (outside the post editor).
echo do_shortcode('');
… is all you need to run a shortcode (code taken from the Codex).
Your request that the shortcode be inserted “somewhere between the page title & the actual contents of that page” is not precise enough to allow for a good answer. There are only a few somewhat reliable theme hooks and none of them are really “between” the title and the content. The best you could do is probably to try to insert the shortcode immediately before the content.
function inject_shortcode_wpse_104354($content) {
return '[shortcodename]'.$content;
}
add_filter('the_content','inject_shortcode_wpse_104354');
The template you mention– “Full Width Page Template (page-full.php)”– is part of your theme. Not knowing what theme, no comment is possible.
Per a comment:
OK, the shortcode is like this: So, when I paste this to
the WYSIWYG, I get the date (when I see preview the page).
That is not a shortcode. Shortcodes use square brackets. You have a plugin that has cooked up its own shortcode-like functionality, or your theme has done so.
You can try…
echo apply_filters('the_content','<!--date-->');
… where you want the date pseudo-shortcode to execute.
But I think a better approach is to find the function that processes <!--date--> and try to use that directly. I am afraid I don’t where to look for that exactly, but you should have some idea of what plugin is doing this.