Your symptoms sound sort of like what happened to us. But our heat exchanger is mounted under the instrument panel near the driver where we could figure out that it leaked, once we went: duh, how come the carpet keeps getting soaked.
My heater core sprang a leak once and another time the hose kinda rotted where it was clamped to the core. You can buy a short little pipe, take the two hoses off the heater and clamp them to the pipe to bypass the core. I can't quite remember but the standard garden hose connector kit might even have worked. When the hose rotted, I just cut off a half inch or so and clamped it back on the heater - there probably is enough slack hose to do this. Used some kind of silicon stuff to really seal the connection.
I would not recommend trying to patch the core if that's where the leak is. Take the old one into an auto parts store and have them match it up with the right replacement from the catalog. I think the right core for a two hose Heatercraft is something like the core used for a 1977 Lincoln, but it could be different for yours. Cost about 55 bucks for a new one for us.