help electrical problem!

The Flashers , running lights and headlights operate with the key off. it may be that your turn signal switch is grounding in the column when you operate the lever.( During the following test, always walk around the car to verify all lights are working, if they aren't, look for a bad bulb.) Leave the key off, put on the flashers. if they work ok, then turn on the running lights. if they work ok ,turn on the headlights. if they work ok, turn on the key, turn off the flashers(the two won't work at the same time)and turn on the right turn signal. if the flashers flash both left and right, you have solved your problem.
Remember, ON a mopar, everything attached to the dash is hot and the body is a ground. Also, The hot on a mopar runs through the turn signal switch all the time, even on the late 60's cars.
I hope this helps.

Here's the thing

If the TS switch is "grounding" it should be either blowing fuses or melting wiring

The head/ park lights SHOULD not have one thing to do with signals except for

grounds

a wiring fault

I have NO idea what you mean by saying "the hot runs through the turn signal switch"

The ONLY "hot" wire to the TS switch

in pre 70 models is the OUTPUT (switched) side of the brake light switch

In 70 and later, the 4x hazard flasher power is "hot" to the switch.

(By the way guy using the word "flashers" is misleading. Are "we" referring to HAZARDS (4 way) or turn signals or both?

On cars built before 70, the hazard flasher and switch IF EQUIPPED was a separate dash mounted switch.