It seems I have anyconnect working to an extent. After I make the connection to our network with Anyconnect, it will not allow already mapped drives to work again.
What exactly am I missing in the setup?
1) I assume you have split tunneling configured and your home subnet is different to the office internal subnet.
2) If point 1) above is correct, you can try to map drive via its ip address.
3) If point 2) works, then it would be a dns/wins issue locally to your home network. Maybe you can add to your localhost file for that specific ip address and it's FQDN mapping.
How is the drive being mapped? Via ip address or name? Do you have dns or wins configured for your anyconnect group-policy? Can you ping the server (by name/by IP)?
Are you able to access other servers in the same subnet as the file server?