Greetings From Arench!
It sounds like both. When a head gasket goes bad and is not repaired immediately the radiator and some times heater core then goes bad from being over pressurized by the leaking head gasket.
I would not replace the radiator until after the head gaskets are done.
If you want to limp along you might want to try a can of alumaseal to try and stop it up. It works sometimes for a short time.