Gas and Oil Home Heating Furnaces - air handler unit keeps blowing

I have a Trane air handler unit. When I turn the thermostat off and have it in the auto position the unit continues to blow. My unit is electric. There are two sets of breakers on the front of the unit. I have to turn it off there to get it to stop blowing. What should I do?
How can I solve this problem myself.

Most likely, your fan relay is sticking. If you open the face of the unit, the is likely a small box that the blower motor leads go into. If you rap sharply on that box with a screwdriver handle the fan will probably stop. That's generally a sign of an aging relay. It's a relatively inexpensive item, you just have to make sure that you keep track of what wires were on which terminal as you swap the relay out.

There are other things (skinned wires from the thermostat, a bad thermostat, etc.), but that's what I'd check first.