I am not sure if the emergency flashers go through the same circuit as the regular turn signals, but there's a turn signal relay that actually does the flashing part for the signals, if that is bad, then you wont get turn signals. I think autozone has them for three bucks. I would check that. I dont know if there is a dedicated fuse off hand that is on that turn signal circiut, but maybe take a look and see if any are blown. In my 66 coupe, that turn signal relay was dangleing down under the dash by my left foot, I just tucked it back up there when I replaced mine, I couldent really see where it was originally. Good luck.
The flasher should be right behind the ash try. Since you have a 68 it has a Turn Signal flasher, and an Emergancy Flasher. You need to find the turn signal one which Is the one behind the ash tray or around it. It is a circular usually grey thing. It plugs into a connector and I believe it has 2 wires. Cost about 2-3 bucks at autozone.
The E-flasher and the regular flasher work in parallel: either can drive the circuitry. Since the E-flasher works, that suggests that most of your wiring is fine. I suggest removing the regular flasher (usually located to the left and above the brake pedal with the e-flasher unit. If your turn signals now work, replace the flasher.
The switch is actually located inside of your steering column, its called the turn signal cam switch or assembly. There may be a problem there with loose of broken wires, but I would check that last. The flasher relay is a round thing maybe 1 inch in diameter, that plugs into a two or three prong wire harness, its the thing you actually hear clicking when your signals are on. like I said, I found mine dangleing down by my left foot under the dash, you'll probably have to get a flashlight and poke around under there to find yours. Let us know how it works out.