Phantom, cars from 1960 didn't have alternators, they had generators. One must keep the age of the vehicle in mind.
Winums, you need to check the voltage under load. Problem is you will need to take the car to the shop for the testing.
"Some" battery shops are still willing to do this for no charge but be warned that some think they have a sucker and will give you a false reading on the tests.
I had a battery shop and world famous Sears automotive both give false readings on the same day. Neither had noticed that the ground strap from the battery to car body was no longer attcahed to the body and without ground the battery won't charge. Both wanted to sell me a new battery and regulator. If I would have returned the next day they would have charged for something else as they were making money by guessing.
Some people, like myself, stillown the equipment needed to check cars pre alt. and electronic ignition. On my set up (Sears) there is a set up that attaches between the generator and equipment which has a method of adding the needed "load" to obtain the correct reading. I haven't looked at the recent home mechanic's equipment to see how and what has changed over the last 20 years so I won't suggest you go out and by said equipment.
But on the other hand, since your car still has a gen. and points it might be worth the price of having a shop testing the battery and charging system.
I will suggest that you add a simple VOLT meter under the dash to keep track of the battery's actual voltage/charging/not charging. AMP meters are supposed to do the same job but on the last three NEW Fords I have purchased none of the AMP gauges worked! On all three I have had dead batteries which would have been noticed/corrected if the AMP gauge worked.
I NEVER trusted Ford's "GEN" warning light either. I've seen way too many Falcons in which the lights had burned out bulbs because the driver didn't notice the light didn't come on when starting the car ("fail safe" design). | 
