My web team let me know that cookies are domain-specific.
I hope they also gave their 2 weeks' notice out of shame.
From the standpoint of cookies, www.example.com and go.example.com (and whatever.example.com) are not different domains. Cookies can be shared among websites using those domains without a problem, as long as the cookie is set to the uppermost private level, .example.com.
The domains are different origins, but origins are not used to decide cookie permissions. Cookies have a much more permissive security model. Your team may have been thinking of cross-origin XHR (Ajax) which does use origins.
Cookies cannot be shared across www.example.com and go.example.org. Those do not share a common private domain suffix.