Submitting Custom Post Types with custom fields from Front-end form

I don’t understand why you registered a shortcode if you aren’t using it. Your custom-form.php‘s code can be:

<?php
    /**
     * Template Name: My Custom Form
     */
    get_header();

    echo do_shortcode('[custom_posts]');

    get_footer();
?>

Now, back to your question, how to save and/or get the custom field value. Well, your method of saving the custom field (meta) value is totally incorrect. I wonder what you got in SERPs when you searched for a related query, or did you even search?

To save post_meta value, first get ID of the newly created post and the save the custom field value. To make the answer shorter, I’ll include the relevant code only

$post = array(
    'post_title' => wp_strip_all_tags( $title ),
    'post_content' => $description,
    'post_category' => $_POST['cat'],
    'tags_input' => $tags,
    'post_status' => 'publish', 
    'post_type' => 'custom_posts'
);
$post_id = wp_insert_post($post);

add_post_meta( $post_id, 'custom_field_one', $custom_field_one, false );

You can also do this

$post = array(
    'post_title' => wp_strip_all_tags( $title ),
    'post_content' => $description,
    'post_category' => $_POST['cat'],
    'tags_input' => $tags,
    'post_status' => 'publish', 
    'post_type' => 'custom_posts',
    'meta_input' => array(
        'custom_field_one' => $custom_field_one
    )
);
$post_id = wp_insert_post($post);

Disclaimer: I am not sure about the rest of the code but only assisted with what was asked.

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