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.