There is usually a built in delay to keep the fan running to exhaust excesses heat. An easy way to check the thermostat is to simply disconnect one of the thermostats wires to turn it off and see what happens.
Some thermostats have an additional ON-Off switch. Any inexpensive household thermostat at the local hardware store will usually do the trick as a replacement.
But, keep in mind that you are working with a 40+ year old gas furnace
and I suggest that, at the very least, you have a furnace tech check it out for operation, safety and any possibility of leaks
. They can kill.