Since the domain has changed, you will have to register the new domain to google captcha. It will then give you a new set of site and secret key that you will have to configure in the captcha plugin.

Since you are not able to login to your WordPress dashboard because of the error you will have to do it outside of WordPress. The goal is to delete the plugin folder of your website.

(1) Control Panel

Most shared hosting websites have control panel where you can access your website’s file and folders. Navigate through wp-content > plugins and find the recaptcha folder. You will have to delete so that when you login to the WordPress admin panel the recatpcha validation will no longer appear.

(2) FTP client such File Zilla

If you are using cloud hosting providers such as Digital Ocean. You will need to connect through File Zilla. There are tons tutorials out there how to use it but once you are connected simply navigate through wp-content > plugin and again delete the recaptcha plugin folder.

In either way, once you have login through your WP dashboard. You will have to install the plugin and configure your new API keys.