Note that your @me.com email address is nothing more than an alias to whatever @iCloud.com you created for it before the entire MobileMe service was turned off back in 2012. So changing your AppleID used with that iCloud to the alias @me.com address really changes nothing at all for that iCloud account. It’s still the exact same account, just using the alias as the primary login ID instead of the previous (primary) email address.
E.g. I use a third party email address as my AppleID (it’s been my AppleID for ~20 years so predates @mac.com, @me.com and icloud), but I can login to my iCloud account using that address, my @me.com address or my @icloud.com address since all three emails are assigned to the same iCloud account. The one listed in my AppleID as “primary” does not affect also using the other two as aliases for the same account.
iCloud: About your @icloud.com, @me.com, and @mac.com email addresses - Apple Support