I remember replacing the heater core on my '85 (TPI) many years ago. Are your passenger side carpets wet?
1) Have you got a Haynes manual for the car that has the fault problems/likely causes listed in it? Can probably find those on the internet now?
2) Have you checked if the car has any fault codes? Can use the bent paper clip trick in the ODB (?) port below the steering the wheel to identify those.
3) At what temperature are the cooling fans coming on? As per factory spec? Maybe (just) a Thermostat problem?
I'm not a mechanic, but hope that helps?