Some time ago old google sites could be connected with my custom domain. Now when I try to add my domain to google site page I get an error to verify my domain with Google. However the instructions points to G Suite which is not free https://support.google.com/a/answer/183895?hl=pl
Is custom domain not free anymore for google sites?
According to what I found online you should still able to make your old google site have a custom domain for free, and then transfer it to the new google sites if needed. Here's the link where I found the answer.
Hope this helps!