Check the two flasher modules in the fuse block
Probably the flasher unit needs to be replaced. This is an electronic switch which usually mounts underneath the dashboard.
I would check the running lights for one with a short. I had the same sort of thing with my trailer and I spent way to much time troubleshooting the brake and signal wires. Turns out a running light bulb connector was grounding out on the trailer and messing it all up.
On the 1979 B200 Sportsman V8 if the brake pedal makes all signal lights come on and hazards and flashers do not work on their own, it could be the flasher, but the culprit could also be a burned out bulb or a short in the wiring.
Try your fuses
I have a chev cavlier 1995 and the running lights do no work---on the dash the green light keeps blinking----what should I do?
The whole turn signal arm unit needs to be replaced
I have run into the same thing with this kind of car. On the steering column at the top they have a switch that will turn the running lights on or turn them off. check the switch you may have hit it and turn it on.
to differentiate it from a brake light or running lights. Its the only one that BLINKS at a regular rate when looking at it from the back.
The only thing that will work with key OFF is the brake lights,head lights and emergency flashers. Dome light will also work.
The only thing common to all those lights (but not to the center brake light) is the hazard switch. I'd check that next. On my 96 Century the hazards override the signals and brake lights meaning that the signals and brake lights must be wired into the hazard switch somehow.
the turn signal switch is probably bad
If all the running lights (front & back) are not working I'd start by checking the fuses then onto the headlight switch