AJAX form doesn't use ajax until cache clear

Question

I'm defining an ajax form in my custom module (#ajax for submit). It is shown in custom block. But it works fine only on cache clear (on Drupal clear all caches or on ctrl+r in chrome). If I just use a link in navbar to navigate to another page or insert an url to browser my ajax callback stops being used. If I ctrl+r it starts working. I thought it's a cache issue but any cache disabling (form or block) doesn't take an effect. Please help.

UPD:

Here is the code for the form:

public function buildForm(array $form, FormStateInterface $form_state) {

    // Add a wrapper div that will be used by the Form API to update the form using AJAX
    $form['#prefix'] = '<div id="my_ajax_form">';
    $form['#suffix'] = '</div>';
    $form['#disable_inline_form_errors_summary'] = TRUE;
    $form['#cache'] = ['max-age' => 0];
    $form['#cache']['contexts'][] = 'session';

    $form['left'] = [
        '#type' => 'fieldset',
        '#prefix' => '<div class="group-left">',
        '#suffix' => '</div>',
    ];
    $form['left']['image'] = [
        '#type' => 'item',
        '#markup' => '<img src="some img url" 
            class="my-form-image" alt="image">',
    ];

    $form['right'] = [
        '#type' => 'fieldset',
        '#prefix' => '<div class="group-right">',
        '#suffix' => '</div>',
    ];
    $form['right']['headline'] = [
        '#type' => 'item',
        '#markup' => $this->t('some text1'),
    ];
    $form['right']['description'] = [
        '#type' => 'item',
        '#markup' => $this->t('some text2'),
    ];
    $form['right']['first_name'] = [
        '#type' => 'textfield',
        '#title' => $this->t('First Name'),
    ];
    $form['right']['last_name'] = [
        '#type' => 'textfield',
        '#title' => $this->t('Last Name'),
    ];
    $form['right']['email'] = [
        '#type' => 'email',
        '#title' => $this->t('Email address'),
    ];
    $form['right']['cta'] = [
        '#type' => 'item',
        '#markup' => $this->t('some text3'),
    ];
    $form['right']['actions'] = [
        '#type' => 'actions',
    ];
    // Add a submit button that handles the submission of the form.
    $form['right']['actions']['submit'] = [
        '#type' => 'submit',
        '#value' => $this->t('SUBMIT'),
        '#attributes' => [
            'class' => [
                'myCustomBehavior',
            ]
        ],
        '#ajax' => [
            'wrapper' => 'my_ajax_form',
            'callback' => '::submitCallback',
            'event' => 'click',
            'progress' => [
                'type' => 'none',
            ],
        ],
    ];

    return $form;
}

/**
 * Validate the title and the checkbox of the form
 *
 * @param array $form
 * @param \Drupal\Core\Form\FormStateInterface $form_state
 *
 */
public function validateForm(array &$form, FormStateInterface $form_state) {
    parent::validateForm($form, $form_state);

    $first_name = $form_state->getValue('first_name');
    if (empty($first_name)) {
        // Set an error for the form element with a key of "first_name".
        $form_state->setErrorByName('first_name', $this->t('Please enter your first name'));
    }

    $last_name = $form_state->getValue('last_name');
    if (empty($last_name)){
        // Set an error for the form element with a key of "last_name".
        $form_state->setErrorByName('last_name', $this->t('Please enter your last name'));
    }

    $email = $form_state->getValue('email');
    if (empty($email)){
        // Set an error for the form element with a key of "email".
        $form_state->setErrorByName('email', $this->t('Please enter your email'));
    }
    else {
        $connection = \Drupal::database();
        $query = $connection->select('my_table', 'm')
            ->condition('m.email', $email, '=')
            ->fields('m', ['email']);
        $count = $query->countQuery()->execute()->fetchField();
        if (0 != $count) {
            $form_state->setErrorByName('email', $this->t('This email already exists.'));
        }
    }
}

/**
 * Form submission handler.
 *
 * @param array $form
 *   An associative array containing the structure of the form.
 * @param \Drupal\Core\Form\FormStateInterface $form_state
 *   The current state of the form.
 */
public function submitForm(array &$form, FormStateInterface $form_state) {
    $field = $form_state->getValues();
    $field_arr = [
        'first_name' => $field['first_name'],
        'last_name' => $field['last_name'],
        'email' => $field['email'],
    ];
    $query = \Drupal::database();
    $query->insert('my_table')
        ->fields($field_arr)
        ->execute();
    $form_state->setRebuild(TRUE);
}

/**
 *  Ajax callback for the form submit
 */
public function submitCallback(array $form, FormStateInterface $form_state) {
    $response = new AjaxResponse();
    if (!$form_state->hasAnyErrors()) {
        $form['right']['#prefix'] = '<div class="group-right group-right-processed">';
        $form['right']['headline']['#markup'] = $this->t('<h1 class="my-form-headline">thank you</h1>');
        $form['right']['description']['#markup'] = $this->t('some text4');
        unset($form['right']['first_name'], $form['right']['last_name'], $form['right']['email'], $form['right']['cta'], $form['right']['actions']);
        $response->addCommand(new ReplaceCommand(NULL , $form));

        return $response;
    }
    else {
        $form_state->setRebuild();
        return $form;
    }
}

Here is the code for the block:

public function build() {
    $form = \Drupal::formBuilder()->getForm('Drupal\my_module\Form\MyForm');
    return $form;
}

public function getCacheMaxAge() {
    return 0;
}

Answer 1

Actually, @Kevin was almost right. There was the search form that does not use AJAX but somehow Drupal gave the same id to both submit buttons (it was edit-submit--2. Is it a Drupal core bug?). So I had to gave custom id to my submit button and now it works fine.