After I light the pilot it will remain on until I switch the unit to heat. At that point the heater fires up and brings the temp up to the desired temp. but the blower continues to blow way after the temp drops in the house below the desired temp.. Some time during this continuous run of the air handler, the pilot light has gone out. So what are the possibilities?

You could have a defective heat exchanger. You could have a defective fan control. OR both