Have you taken the switch out? I would check the springs to see if they can be replaced and use some electronic tuner cleaner.
No, I'm just doing some research before I get started. I just read the service manual, and it has a great explanation. Guess I'll check the spring and adjustment first.
Unless the switch is actually broken, my suspicion is that there is a spring tension or lube or both issue.
Maybe apply just some spray electronic cleaner first. Friction from 50 plus years of contaminants and loss of the original spring tension would be my first checks. First, post a picture for comment on lubricant. Do not use an oil base lubricant. A greaseless lubricant is made, but try cleaning it first.
Haven't torn it apart yet, but did take it for a ride this morning. The right hand turn works. I can see and hear the left hand return to the center position, but signal continues to flash. If you switch to the right and back to center it will stop.
I'm not sure about a 1961 car, but the 70's cars had these little springs(circled), one left and one right. They are replaceable. I remember I got them at a dealer, he had a whole box of them back then. Here they are on Ebay, https://www.ebay.com/sch/i.html?_fr...&_nkw=turn+signal+cancelling+springs&_sacat=0