Sorry folks, I emailed Greg L (too many Greg's?!) a few days ago to tell him I found a solution to my problem where I can't connect with Windows Authentication.
Greg W's suggestion is almost what I did: I created a domain Group called 'SQL Administrators', put myself in it and then added that group to Instance > Security > Logins and gave it the sysadmin role. I thought I'd have to tick all of the User Mappings or add the group to each database, but that wasn't necessary. Simply creating the sysadmin Login gave me implicit control over all databases. I'm not sure why I didn't have any problems on my old domain, but it's destroyed and I can't look back to see why. Greg K