When you adjust the thermostat it will heat to where it should be but then the fan will just keep running without the heater even being on and just keep blowing out cold air.
If the furnace blower is operating and the fan is in vehicle position...this suggests there's a call for cooling and could be sending 24 volts on your contactor outdoors at condensing unit. If there is 24v at coil of contactor and 240volt at line with contator closed...then [if neither compressor or condensor fan are operating]: a million. defective contactor or 2. defective run capicitor you want to discover once you've both line voltage [240v] and administration voltage [24v] at outside unit before figuring out this as a difficulty. HVAC service Tech :)
Ok I'm sure you made sure your thermostat wasn't in the fan on position. Beyond that there is basically only one thing that could be doing this on a gas furnace the fan is controlled by a plenum switch that brings the fan on when the heat exchanger heats up and turns it off when the exchanger cools it sounds like this guy is sticking on. On an electric heat unit it could be a fan relay causing the issue. Either way something you can try is to smack the unit with your hand or a rubber mallet, if the fan turns off then it is obviously one of those.
Here you go, I hope this helps. Check the trouble code on the board, which newer furnaces have. Check thermostat and make sure the fan switch is in auto. Otherwise it would be a limit switch in your furnace (around the combustion chamber could be 3 and on the blower). Limits will often default to fan on continous when they are tripped. Fan/limits will do the same. If you reset one of the limits and it trips again post limit tripping)
It may have a timer on the fan that keeps it running for a period of time after it stops heating/cooling, some are adjustable and some are pre set. I would advice you to contact the manufacture and as for guidance or contact a suppler of installer to look at it for you. Please do not attempt to correct it your self because you can catch Legionnaire's disease from an air conditioner system if it is not maintained and cleaned correctly.