Thanks to Dan and Chuck for some really good suggestions. Dan, I looked at the schematic and saw that one of the wires from the headlight switch does go to the turn signal switch, but it appears to be connected externally to another wire, and isn't affected by the headlight switch internals. I flipped the lights on in different positions and tried the turn signals anyway, but still they were solid on, no blinking.
Chuck's suggestion for looking at one side of the flasher while the other side was plugged into power made sense to me, so I tried that and found a very strange result. I made up a couple of jumper wires to run from the connector under the dash to the flashers just so I could monitor voltage input. Both the emergency flasher and the turn signal flasher were getting 12.5V, but the emergency flasher flashed on the output side, and the turn signal flasher did not. Scratched my head over this for a while, but finally as I was staring under the dash at the two flasher connectors I figured it out. I had gone through the circuit diagram numerous times and knew which color wires got power; on the emergency flasher circuit it was a red with white stripe wire, and on the turn signal flasher it was a blue wire. Of course the flashers can only go onto the connector one way, and I saw that the voltage input connection on one connector was on the left terminal, and the voltage input connection on the other flasher was on the right terminal!
I connected my jumper wires to the turn signal switch flasher, and swapped the wires at the connector, and the turn signals flashed! So, problem solved; but...
But then I got to thinking about the factory style flashers, which were essentially just a switch that opened and closed. In that case it shouldn't matter which terminal got the power, the switch contact would still heat up and force a flash. I had gotten a replacement flasher from Napa a couple days ago, and that was the one I was using. I unplugged it and replaced it with the electronic flasher from Ron Francis Wiring, and checked to see that the turn signals still flashed. Then I pulled the top off the flasher from Napa; sure enough, it was electronics inside. So the lesson I'm taking away from this is that the electronic flashers are polarity sensitive, but the factory style flashers probably are not, given the way the Cobra is wired.
In any case, now the turn signals work. Of course, there is more work to do. I took the car out for its first test drive on Thursday, and after idling in the driveway at 180 for 15 minutes, it went up to 240 in one lousy mile of driving! The radiator looks like junk, and it is probably is. New radiator and electric fans are coming on Monday, so hopefully I'll finally get some miles on this thing next week. Thanks again for all the suggestions from the forum - Jay