Once you know how all your site-generated mail is being sent, you can set up your DNS records to help validate it.
AWeber is not using WordPress or your server. The SPF record you attached to your domain for use with AWeber tells mailservers that email associated with your domain may actually originate from aweber.com.
Transactional emails generated by WordPress and plugins like WooCommerce and Gravity Forms are a different story. By default, those emails are created by PHP mail() through WordPress’s wp_mail() wrapper function so they originate from your webserver unless you or your host have set up an SMTP alternative like SendGrid or Mailgun. (Most Managed WordPress hosting works this way.)
Using a third-party mail service is likely to have the best results, but you can send transactional emails from the webserver too with the right configuration and throttling. The biggest and most common barrier to deliverability for emails sent by the webserver may be a mismatch between the return-path and “from” addresses in the WordPress-generated mail headers. These should generally be the same address, and it should really exist.