[VetteVet]

When you turn on the headlights you are getting a voltage drop to the rest of the lighting circuit. If you have poor grounds then the turn signals will ground through the path of least resistance, usually the dash cluster. I would ground the cluster with a separate ground from the tin to the cab chassis. Also the headlight grounds on the core support need to have good clean connections.
One or more of the turn signal housings could have a poor ground. I would suspect the front turn or marker lights since they power the turn indicators in the cluster.