How to add a custom form to a plugin’s setting page

The fourth parameter of add_menu_page is the menu slug.

Hence the form action should start with: admin.php?page=vsp_settings_page, if you want to stay on said page.

If your form is correctly written, it should populate the query string in the URL.
You should have probably posted all your form’s markup.
Is there something along the lines of

<input type="text" id="offset" name="offset" value=""/>

in it?