Adapt PHP form action for WordPress?

I think your action URL is wrong. Did you try with

<?php echo plugins_url( 'listrak-newsletter-api.php', __FILE__ ); ?>

like this way:

echo '<form action="' . plugins_url( 'listrak-newsletter-api.php', __FILE__ ) . '" method="post">';

in your code?

Anyway I still recommend you to use WP AJAX to do this. Read here for more information.

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