The system continues to run even when in the "off" position, no matter if it's set on heat or cool, and no matter how or low I turn the thermostat. I have the thermostat set on "off" now, and the fan continues to run; it's set on auto. There is no off position for the fan, only for the heat/cool settings. The air being blown out is neutral; the heat worked last night; I discovered the malfunction when I tried to turn the heat down, then finally off. Only the fan continues to run. I'm afraid to try the air; I'm afraid I might may things work, and living in the deep south, we'll need the cool air next weeK! Any ideas, as this a weekend, and my husband is out of town?
Thank you for any ideas.

It sounds like your fan limit switch is stuck try pounding on the side of the furnace with your fist see if it stops if it does a simple adjustment on the fan limit will take care of it

For starters, it won't hurt much to let the fan run a few days. Some let it run all the time. In newer heating systems the fan isn't controlled by a limit switch. It is possible the red and green wires to the thermostat are shorted, or the thermostat is bad. Most likely the relay on the control board is stuck. Finding it and giving it a rap might stop the fan. That should confirm you have to put up the money for a new board. In some systems, the fan will run if it loses control voltage. That is a easy fix in most cases by replacing the fuse. You can find the details you need in the sticky at https://www.askmehelpdesk.com/heating-air-conditioning/heating-cooling-maintenance-repair-troubleshooting-frequently-asked-questions-58313.html

I have a similar problem with my Lennox that hasn't been fixed yet. What I do is throw the breaker switch for the fan in the main breaker box for the house and it resets and turns off. Try that for a temp fix.