From what table occurrence did you select the FullName field?
If your relationship reads like this:
Then adding Carers::FullName to your layout should display the full name of the selected Carer record.
Ah. Solved it. I actually had a relationship both ways.
'Carers' ID --> 'Contacts' CarerIDFk & 'Contacts' ID --> 'Carers' ContactIDFk
When I removed the second, the full name displays now.
Second question is how can I have a button so that when I click on the Carer name it goes to that profile in the Carer layout? If I make that field a button then a) it takes me to the first person in the Carer layout, not that specific person whose name I clicked on and b) obviously if i try to add a Carer to a Contact I can't click the field to get the drop-down list of names -- because it's a button.
Look up Go To Related Records in FIleMaker help. You can set your button to use this method to change layouts and also bring up the related record or records linked to the current record on the layout where you placed that button.
Thanks very much, PhilModJunk. All sorted now!