Yes, the DHCP is set up with DNS correctly as far as I can tell. All devices that use DHCP get the DNS server details correctly (the DNS Server is an Windows 2012 Machine) and can resolve external host names.
Our setup is currently this.
ASA 5505 is the default gateway, running a DHCP server. We have a DNS server @ 22.214.171.124 (Windows 2012) and the DHCP server has this IP set as DNS Server. It also has 'Update DNS Server' with 'Update both records' and 'Override client' checked.
Most devices on the network are using DHCP, they get the DNS server and can resolve external host names correctly. A lot of devices do have static IP's and have the same DNS server configured, however no internal host names get resolved.