97 Z28 - When turning left or right blinkers on, the turn signal will click for several seconds then stop and the light will fast flash (both inside and outside) sometimes the clicking kicks back in and the signal behaves normally.

All the lights are fine.

Bad blinker socket? (I'm puzzled by the fast flash... usually a bad blinker just causes the light to either stick on or off...)

 

Fast flash is usually caused by high amps. This sometimes happens when one of the lights burns out, the other will flash faster. I would check the sockets, wires and ground for corrosion. Also it might be something in the steering column, turn signal mechanism. The flasher itself might be getting "tired". Check the easy stuff first.