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The flaher is probably burned out. The odds of 1 bulb on each side of the car burning out are slim. On my 87 I think the flasher is behind the trim panel that covers the radio. Change both flashers to be sure. They are dirt cheep
First check your front and rear turn signal lamps. If any are burned out, the blinker will not cycle because it does so with lamp current. Low lamp current will slow the blinker and if too low, the blinker will not operate.
Try turning on your hazards- if the hazards work fine, then your blinker module probably went. A new one can be nabbed at PepBoys or similar for a few bucks. It is a bit of a PITA to get at, but it fixed my problems (was identical to yours.. blinkers were solid on, for both left and right.. but the hazards worked perfectly).
If you need to replace it, do as follows:
1) Disconnect negative terminal from battery
2) Drop carpeted kick panel on drivers side (under dash)
3) The blinker module should be mounted up, behind (towards the back of the car) the hood latch. It is real tight up there. I used a screwdriver to pop it out of its little holder.
4) remove the plug from the old one, pop in the new one, push it into the clip that holds it in place, and button everything back up.
Took me about an hour to do.. 40 minutes of which, was looking for the damn thing.. (laying on my back, under the drivers side dash!!)