Wow. It definitely sounds like vapour lock.

BOY am I kidding there!
This sounds very fixable. I'm guessing you'll dive under the dash and see something obvious around your ignition switch or your fuse panel. Check for warmth. Somewhere along the line, asking for current draw from one of the supporting systems is either overtaxing a bad connection, or something like that.
I'd throw a multimeter on the +12V connection on the ignition switch and "make" the car die. See what voltage reading you get. If you get something funny, work back towards the battery. If not, work forwards towards the ignition coil or the ignition box. Somewhere along the line you're dropping the voltage below the point where the ignition system will run the engine.
Both the things you found involve the signal/marker lights, though. Is there anything else that'll kill it? Not a whole lot of electrical components in that car to begin with... heater fan maybe? At least you can test this stuff in the garage now!
Did the poor running before correspond to the hazards being on, perhaps?
The blinkers all shine brightly? Marker lights & headlights work fine as well? Other than shutting off the car, all systems are a go? When you pull the fuse for the headlights/signal lights, does the problem continue? Brake lights don't cause the problem, obviously?
Another edit - you might not see the problem after you first fire up the car, as the alternator is putting out big voltage to try and recharge the battery. Also, it might take some time for the particular bad connection to heat up and open on demand.