I'm not sure what you are saying in your original post. Do the blinkers work normally when the headlights are off?
Turning on the lights increases the load on the system, so yes, they can affect the blinkers.
As Ed suggested, increasing the engine RPM allows the charging system to compensate for the increase in load. If the charging system cannot keep up, there may be a problem with it. (generator, belt,, regulator, battery, wiring).
I'd start by testing the charging system. Get a VOM and see what the voltage is at the battery with the engine running. Does the voltage drop when you turn on the lights? (more that just momentarily)
