1996 Honda Accord

Last night I was driving with my heater on. All of a sudden the heat went away, but the air was still blowing through the vents. I tried pushing buttons, turning knobs, etc. Nothing worked. Then I noticed that my car was starting to overheat. Today, same thing. I checked my radiator and saw it was low on fluid so I filled it up and now the car isn't running hot per se, but the fans are still running for an extended period of time once the car is turned off as if it is running hot and the heater still is blowing cool air. Is this a thermostat problem and if not, do you have any ideas what it might be instead?