I have an older, central H&A system with a fan that won't stop running. Heat & air both work fine, but once the house gets to the right temp, the unit will cycle off but the fan keeps running regardless of whether the fan is switched to "on" or "auto". Many times the fan will stay on for up to 4 or 5 hours after I turn everything off at the thermostat. This is an all-in-one unit outside (no separate air handler inside the house). Any ideas on what's wrong??

There are a few things that will make a difference in what I say. Is the unit a heat pump, or gas furnace? If a heat pump the speeds for cool are the same speed as heat, so the relay that runs the fan(fan relay) controls the fans operation, either a separate 24v relay or a relay as part of the circuit board, tap the relay or board to see if they shut off the fan, it can be stuck.

But before we get that far, if you have a gas furnace as most do, the low speed is the heat speed and is controlled via temperature delay, that is located in the high limit safety, located above the burners generally, the limit and low speed fan switch open and close via the little clock like dial w/temps marked which rotates via a spiral bi metal that expands and contracts and that opens and closes the switch for the low speed fan operation. The high speed fan or cool speed is run via a fan relay, and nothing to do with any temperature delay as does the low speed (heat speed) So if its doing it with the heat only and not cool, it's the temp delay, if it does it all the time, you must have 1 speed for both modes? A little more info required , but bang on the unit and see if it stops , that says a relays stuck.