Sometimes the flashers are such that the load of only one bulb is not enough of a load to get it to flash, as the bimetal strip inside the flasher doesn't get warm enough.
Suggestion: get the other bulbs to work first to see if that gets the flasher to flash, OR get an electronic flasher which will work on only one incandescent bulb (most electronic flashers are that way) to get the flashing to work.
It's possible it's the turn signal switch itself, but from what you describe, I'd look to solve the other bulb non-illumination probs first. Try applying 12V to the various terminals on the dash harness (that connects to the turn signal switch) while the turn signal switch is disconnected from the harness connector. Do one terminal at a time that leads to the various corner lights of the car (right rear, right front, left rear, left front). If each light works (left rear, right rear, etc.), the switch or flasher may be suspect.
If each light doesn't work, solve that first. Typically grounds (are lack thereof) are the first things to check. Use the wiring diagram of the FSM to determine which wire is which.
