Check your grounds on the core support. You should also check the bolts attaching the core support to the passenger's fender, and the battery ground on the fender support If the grounds are bad, the lights will try to find another path for current flow. In your case it is through the turn signals. Checking the battery ground to the engine, and the engine ground to the body.
My brother had a somewhat similar issue years ago with a Plymouth that had a 440 in it. When he tried to start the engine, the solenoid would click, but the starter would not run, although the headlights would come on dimly. It turned out to be the engine block ground. Some "intelligent" engineer had the engine ground go to a bolt that penetrated the water jacket.