Trying to replace the conventional thermostat with the programmable one. Went to HD and the Honeywell ones have chart in back for compatitbility. My system is a heat pump with a heating element for emergency. However, the chart only list AUX but does not list emergency heat. Is AUX the same as Emergency heat?

Earlier this fall, we had a cool streak and didn't use the A/C for a while. After it warmed back up, the A/C wouldn't work. The ducts in the house blow air, but not cool air. The outside unit wouldn't turn on at all. We just went along without A/C because we figured it would cool off again fairly soon. When it came time to use the heater, it heated just fine, but the air blows out of the ducts almost continuously (blowing cool air after the heat shuts off). The outside unit does turn on when the heater is in use. The house stays warm, but cool air is constantly blowing out the ducts and it seems like that would be making the inside temperature cooler than if the air wasn't blowing. Does anyone have an idea where my problem might be? Thanks in advance!