The fan for the AC came on and won't shut off. The outside unit is not running. The thermostat is set on fan off and heat/cool off. It is an old manual thermostat. Is it the thermostat?

It could be the thermostat or a fan blower really.

I just had that problem and seen a couple of replies that said it may be either the fan relay or the thermostat. I fixed it the old way... I tapped on the limit switch. Makes sense because if the temperature is high enough in the chamber(fire box) the limit switch will keep the fan running. In my case, the contacts must have been stuck. It may be worth a try. The limit switch will have a couple of adjustments, remember what they are set at before turning them... better not to try to adjust it. Good Luck!