Yes they do.

Check the grounds, most electrical issues in these old cars can be attributed to corroded ground wiring connections.

Rudi wrote:

Check the grounds, most electrical issues in these old cars can be attributed to corroded ground wiring connections.

I agree.  Grounds often go thru the body of the part to the sheet metal of the car.  I have often sanded both sides to get a good ground.  Metal to metal, with perhaps some dielectric grease for sealant.  In any case always make grounds go metal to metal.
 

If the lights weren't working I'd suspect a ground, or in a newer car where things are switched via ground, but I don't think this is a grounding issue.  The ground for the bulbs is the socket to the taillight housing to the body.  If the ground was bad the lights wouldn't work.  The lights come on via switched power, so something's switching power improperly, or power is crossing circuits it shouldn't. 

It's also not the headlight switch.  The headlight switch simply feeds power to the brake light switch, which then sends power to the bulbs through the t-signal switch.  The brake light switch must be working correctly because with the headlights off the brake lights function normally  (the switch is getting power and isn't stuck on).  The only way the headlight switch can affect the brake light circuit would be for it not to be sending power to the brake light switch.  We know it's doing that.  Otherwise the brake lights and taillights operate off separate circuits.  There's nothing in the headlight switch that could be crossed, etc. and make the brake lights come on, because the power for those lights is breakered by the brake light switch, and without that switch being closed (pressing the pedal) there's no way to turn them on. 

Take out one of the taillight bulbs and test each terminal for power with the headlight switch off and on.  My guess is what you will see is that you have power at both terminals with the headlights on.  You should only have power on one terminal.  The taillights turn on one filament; the brake and t-signals another.  For some reason you are getting power to both filaments with the headlights on. 

I'd check those taillight sockets closely, because the most likely cause would be that there's something bridging those two terminals or the wires leading to them, and that is the place where they are in closest proximity.  When I replaced the taillight harness on my '67 I had to reuse the sockets.  With the harness having been replaced I have to suspect that the last thing you did was what caused the problem, so unless everything worked correctly after the harness was installed and now all of the sudden doesn't; the issue lies with one of the harnesses that was replaced.