I just went through this with my '60 Eureka. You're lucky the car doesn't have a/c because it makes the job take longer. You will have to remove the Bowden control cables from the heater control valve before removing the heater assembly. There is a small access panel in the heater box that you would open to do this.
The heater control valve is attached with nuts to studs on the side of the heater core. A gasket is sandwiched between the valve and the core header. It is definitely possible that the rubber stoppers (or flow controllers) of the valve are leaking and that the core is OK.
Remove the heater box assembly from the car, remove the heater core and valve from the heater box. A good radiator shop will dunk test them both and see where the leak is. There is an outfit in Windsor CT that does a good job of rebuilding the valves. They do a really good job but are a bit expensive. In my case I did not want to ever remove that heater again so I opted for the valve rebuild as well as a new heater core, which was the source of the leak. If I can help any further please email me directly:
hoczyk@yahoo.com. Tom