Since no one has responded after a day, I'll provide my very limited insight.
If you don't want any personal information on your work Mac, then I would avoid signing into your AppleID on the work Mac.
Theoretically it may be possible to use an AppleID with just the Messages app as an example and nothing else, but even that is fraught with dangers & problems. I'm using this as an example because that and my Contacts are the only things I associate my AppleID with on my work Mac....and it is not easy to do. In fact when I migrated to a new Mac and signed into my AppleID.....macOS activated nearly every cloud feature and started to automatically sync my files with iCloud without my permission. If for some reason you need to log back into the AppleID to use that app/service, you must be extremely careful since Apple doesn't provide an easy way to just login to the AppleID for just than one service/app. The only way I found to do it is quit the Messages app in my example & relaunch the Messages app so I am presented with the login prompt for Messages (the last time I had to do this was with macOS Catalina, so I don't know if that is even still possible with later versions since Messages hasn't lost its connection/association with my Apple since I migrated to a new Mac).
As for the Continuity feature....I don't know since I've never used it. From a quick glance at Apple's documentation, I don't think there is any app that you sign into....it sounds like you must sign into the AppleID fully where things may automatically start syncing to the cloud (possibly from the cloud as well).
Apple assumes everyone is all in on using everything associated with their AppleID. Personally I think Apple should default to having all cloud based settings disabled when logging into an AppleID for the first time on a new device, but that is not what occurs.....at least in my own personal experience. I even waited until after I migrated from my old Mac before attempting to log into my AppleID on the new Mac (I think I even tried to just launch the Messages app and log in that way, but I'm not sure now since the migration was such a huge pain to perform.....all the problems were associated with Apple apps, settings & services.....third party apps & settings all migrated smoothly....go figure).
