Search query with filter

[admin]

Thanks to the help of stack overflow I was able to make my filter work(<a href="https://stackoverflow.com/questions/65347618/ajax-filter-for-wordpress">Ajax filter for wordpress</a>) but now I want to add a search input that search the titles of my posts (a custom post type called 'Contratistas').
So I have a few questions:
<ol>
<li>My form for my filter has POST, but the search (i've seen examples) always has GET, so, should I have two separete forms? If so, is there a way for submiting my search without a button? this is my code:</li>
</ol>

<form id="searchform" method="get" action="<?php echo esc_url( home_url( '/' ) ); ?>">
    <input type="text" class="search-field mb-3" name="s" placeholder="Search" value="<?php echo get_search_query(); ?>">
    <input type="submit" value="Search" class="mb-3">
</form>
<form action="<?php echo site_url() ?>/wp-admin/admin-ajax.php" method="POST" id="filter">
    <div class="titulo mb-3">
        <h3>Región</h3>
    </div>
    <?php
            if( $terms = get_terms( array( 'taxonomy' => 'region', 'orderby' => 'name' ) ) ) : 
     
                echo '<select name="filtroRegion"><option value="">Seleccione una región</option>';
                foreach ( $terms as $term ) :
                    echo '<option value="' . $term->term_id . '">' . $term->name . '</option>'; // ID of the category as the value of an option
                endforeach;
                echo '</select>';
            endif;
            
        ?>
        <div class="titulo my-3">
            <h3>Industrias</h3>
        </div>
        <?php   
            if( $terms = get_terms( array( 'taxonomy' => 'industria', 'orderby' => 'name' ) ) ) : 
     
                echo '<select name="filtroIndustria"><option value="">Seleccione una industria</option>';
                foreach ( $terms as $term ) :
                    echo '<option  value="' . $term->term_id . '">' . $term->name . '</option>'; // ID of the category as the value of an option
                endforeach;
                echo '</select>';
            endif;
        ?>
            <button class="my-3 filtrar">Filtrar</button>
            <button><a href="" id="clear">Limpiar filtros</a></button>
            <input type="hidden" name="action" value="myfilter">
</form>

As you can see i have two buttons but i only want the last one ('Filtrar')
<ol start="2">
<li>I have no idea how to implement the search with the filters so that i can put my results in the same fashion as the one i get from the dropdowns.</li>
</ol>
Here is my filter:
<pre class="lang-js prettyprint-override">

$args = array(
    'orderby' => 'date', // we will sort posts by date
    'order' => $_POST['date'], // ASC or DESC
    'post_per_page' => -1,
    'post_type' => 'contratista'
);

if (!empty($_POST['filtroRegion']) || !empty($_POST['filtroIndustria'])) {
    $args['tax_query'] = array();

    if (!empty($_POST['filtroRegion'])) {
        $args['tax_query'][] = array(
            'taxonomy' => 'region',
            'terms' => $_POST['filtroRegion']
        );
    }

    if (!empty($_POST['filtroIndustria'])) {
        $args['tax_query'][] = array(
            'taxonomy' => 'industria',
            'terms' => $_POST['filtroIndustria']
        );
    }

    if (count($args['tax_query']) > 1) {
        $args['tax_query']['relation'] = 'AND';
    }
}

$query = new WP_Query($args);

And my JQuery:

jQuery(function($) {
    $('#filter').submit(function() {
        var filter = $('#filter');
        $.ajax({
            url: filter.attr('action'),
            data: $('#filter :input').filter(function(index, element) { return $(element).val() != ''; }).serialize(), // form data
            type: filter.attr('method'), // POST
            beforeSend: function(xhr) {
                filter.find('.filtrar').text('Procesando...'); // changing the button label
            },
            success: function(data) {
                filter.find('.filtrar').text('Filtrar'); // changing the button label back
                $('#response').html(data); // insert data
            }
        });
        return false;
    }
}

Any help or guidance will be very appreciated. Thanks