Hi Steve,
Your groups are set up correctly. I suspect that the directory takes time to refresh. You have two options as I see it:
Newly created Groups take up to 6 hours to show in the directory – see this Google article for reference. One possible solution is to ask staff to type the email address of the group they wish to email for the first time, then see if it appears the next time they want to use it. I wonder if this forces the directory to refresh…- You would upgrade to Groups for Business, which is a super-charged version of Groups and has many more features. It is more sophisticated but once things are set up it just works.
I’d recommend testing the first step and then let me know how you go.
Rob