Why would the blinkers on a '92 Chevy 1500 not work while the lights and brakes are on?

May want to check your dimmer switch, too - it could just be in the off position. You could have a shorted relay if it only happens when you turn your headlights on. From the description you gave, it sounds as if the instrument cluster has come unplugged. You may be able to reach up behind the cluster and plug it back in, but usually the cluster has to be removed to access the plug.