I don't think the Drake and other similar after market fog lamp kits include a relay but really they should.
Mark, the original wiring harness in these cars has the full electrical load imposed by the headlights going through the headlight switch. While this might have been OK 50 years ago when we only had weak lights it's not really up to the draw of modern halogen or other lights so putting a relay in the system is a common and good idea. There are kits available for this - I got one from forum sponsor Custom Mustangs a few years ago. A relay is just an electrically operated switch really. You mount it close to the lights and the power goes a short distance from the battery (or starter solenoid) to the lights via the relay. A longer and lighter duty wire goes from the relay back to the headlight switch which is then only switching on the relay to complete the circuit between the battery and the lights, rather than having the full headlight draw going all the way from battery to headlight switch and back to the lights. Headlight switches overheat and your lights play up without a relay with a modern headlight fitted.
As Boof suggests, the same issues are at work with fog lights. The Drake kit is really adding to the problem described above by adding the fog lights to the load going through the headlight switch, particularly if you have both headlights and fog lights on together. A relay in the circuit close to the lights themselves would be a good idea, but as I said I don't think these kits usually provide for one so you'd have to modify it. It's not that hard.
You can get a relay from Jaycar or somewhere for only a few bucks. You would need to insert it into the circuit near the lights and run a separate feed from the battery or solenoid to the relay. In this diagram
http://www.next.gr/uploads/39/relay_diagram_02.gif the trigger source would be the fog light switch getting power from the source you found, the device would be your fog lights. You would need to cut the wire in the kit that goes from the lights to the firewall, connect the light end to terminal 87 and the other end to terminal 85, then a separate power feed to terminal 30 and earth to terminal 86. Job done!
Boof suggests a switched power source, i.e. only live with the ignition on, but the headlights use a constant power source which I presume is what you found unless you had the key on while poking around with your test lamp.
These links may or may not be useful
http://www.cjponyparts.com/fog-light-wiring-kit-1965/p/WFWK1/http://www.mustangproject.com/Links/1965-1966%20Mustang%20Fog%20Light%20Kit%20instructions.pdfGood luck.