The indoor fan never stops running. You checked the obvious. The fan switch on the thermostat is in the "auto" position, not the "on" position. You try turning the thermostat to the "off" position. Nothing works, the only way to get the fan to stop is to shut the power off.
Well, if you are going to have a problem with your HVAC equipment, this is the one to have. The heating and air conditioning still works fine. If you have allergies your fan should be running all of time anyway to filter the air more. It really doesn't hurt the motor running all of time. And it usually isn't an expensive repair.
Just like the previous problem there are usually only three things that cause this. A stuck fan relay, shorted thermostat cable, or a bad thermostat.
That is about all there is to this problem. So if your fan doesn't shut off, don't panic - just schedule a service call and have it repaired.