Using a Rewrite URL in a Plugin to Load a New Page in the Template

add_rewrite_rule( $regex, $redirect, $after );

---

Rewrite rules to scripts other than index.php

The $redirect argument works slightly differently when redirecting to a custom PHP script because WordPress delegates these redirects to .htaccess instead of processing them itself. For this reason, querystring variables should be written like $1 instead of $matches[1].

Personally, I would use a rewrite rule pointing to index.php. In this way, the rule will also work outside of Apache.

add_action( 'init', 'se343855_rewrite_rules' );
add_filter( 'query_vars', 'se343855_query_vars' );

function se343855_rewrite_rules()
{
    add_rewrite_rule(
        '^page/form/([^/-]*)-([^/]*)?$',
        //  Or maybe:
        // '^page/form/([^/-]*)(?:-([^/]*))?$',
        //
        'index.php?sign1=$matches[1]&sign2=$matches[2]',
       'top'
    );
}

function se343855_query_vars( $vars )
{
    array_push( $vars, 'sign1', 'sign2' );
    return $vars;
}

Load a custom template if the query variable is set (sign1 or sign2):

add_filter( 'template_include', 'se343855_template_include', 50 );

function se343855_template_include( $template )
{
    $sign1 = get_query_var('sign1', false);
    $sign2 = get_query_var('sign2', false);

    if ( $sign1 !== false || $sign2 !== false ) {
        // if this code is located in plugin file
        $template =  plugin_dir_path(__FILE__) . '/pages/form.php';
    }
    return $template;
}

Use get_query_var() to get value of sign1 or sign2.

In form.php file you can use get_header() and get_footer().