The bulb will have two filaments, and two connections at the base. One for tail lights, the other for brake lights. It grounds around the perimeter of the base. The socket will have two connectors. That is another place to look for a short. The insulation on the wires where they go through the base of the socket may have gone bad. That will cause a direct short. If the short is at the light socket, usually the wire will burn at or near that point, but not always.
Lots of places to look for shorts to ground. But if the fuse does not burn when the running/tail light wire is removed from the switch, the short is somewhere along that wire. With the unit being fairly old, I might start by looking at the wires where they go through the sockets. Age coupled with vibration can cause the insulation to flake off the wires. I replace the sockets or at least the wire and socket base on lights on older cars during restoration. With age, the spring beneath the bulb connector and the socket base goes bad as well.
I will be interested to hear what you find.