This sounds like a poor ground connection. Clean the negative battery terminal and the ground connections from the battery negative to the radiator support or fender, and the battery negative to the engine. Also clean the ground from the firewall to the engine.
It's not enough to just look at them, you have to clean them. If that doesn't solve the problem at least you can tell the shop what you've done so far and save a few bucks that way.
But bad grounds are most often the culprit when you see this kind of behavior.