Submit form to a different PHP file in the same plugin folder

You should not try to do it. Direct access of php files which are not in the root directory might be blocked for security reasons.

what you need to do is register a settings page for your plugin and add into your forms some hidden field that indicates what kind of processing you want to do on the $_POSTed data. Then use the URL of the settings pag as the action url for the form (use admin_url to calculate it).

Side note: always try to avoid relative urls in wordpress forms. It is too easy to end up with the form at a place where the relative url points to nothing.

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