WordPress: URL change and cookies issues

I had a site under alpha.mydomain.tld and when I changed it to www.mydomain.tld I did it trough the Settings of WordPress without changing anything else.

I added the new domains to my nginx-configuration without removing the alpha subdomain and I got a lot of issues with WordPress cookies. By e.g., I logged in to /wp-admin, closed that tab and reopened in a while and I got to re-login, even if I set "Remember Me".

After checking, re-change of URL and things like that I just removed the alpha subdomain from my nginx-configuration and everything got solved. Yes, is stupid, but it happens.

Any comment goes best to my twitter here.