Facing the same issue as Peter_Holdridge - when trying to access the new URL, we get prompted for an account 3 times, then end up on a page that says 'Switch to an account that has permission' - even with a global admin account.
Same issue in incognito mode, except there you only authenticate once, then get redirected to login.microsoftonline.com a few times, ultimately ending up on the same error page.
[UPDATE]
Just noticed that this seems to depend on the tenant you connect to. And it's not specific to the Exchange admin center, it's just the admin.cloud.microsoft site that has this issue for specific tenants.
The tenants where I have this issue consistently produce this error, the tenants where I don't always seem to work, so I suspect some setting within the tenant might be causing this?
Haven't checked them all, but I do know one of the tenants with issues has a conditional access policy restricting admins to specific countries. Could be the 'new' portal just forwards requests to the 'old' portal, but this changes the IP/country of origin?