1997 Ford Taurus A/C overriding heater?

Sounds as heater core issue and Flushing it should resolve the problem

try and do this one more time

disconnect the heater core hoses from engine side
make sure you plug the bypass hose between the two lines
use air and plow out what in the heater core
have the hoses up and use a can of flush
let it sit in for few min as the longer the better
than use a garden hose and compressed air and flush the core
make sure flushing it both ways
hope that work for you
always did for me on the stubborn ones