I have a heat pump. This unit does have emergency heat. I only know this because I have used it. I noticed that the other day the fan stays on with the heat on. The fan switch set to auto and the temp set way below the actual room temperature. I removed my thermostat to examine the wiring and it has E and W2 connected together. Could this be the reason for the fan constantly running?
The heating fan will run for a cool down after the heating elements are turned off. If you listed your Furnace Model and your thermostat model we could check the wiring for you and suggest what you need to do.
But yes it sounds like the Thermostat is wired incorrectly.