Does anyone have an idea as to why the fan on my heater/AC would run
constantly when the thermostat is set to have the fan in "Auto" mode
but not run at all when in "On" mode?

The whole story:

My fan was running all the time regardless of thermostat setting,
including heat and AC off. I disconnected the green wire from the
thermostat and it kept running. From everything I read that indicated
a stuck fan relay. When I looked at the fan really it was cooked,
burned plastic, etc. So I replaced it with the same model. I'm
pretty sure I wired it exactly the same, I took notes, etc. So now I
got the weird situation described above. I picked up a new thermostat
thinking that maybe the burned relay took out the thermostat but it
does the same thing. That makes me believe I did screw up the wiring
but after much reading and checking and rechecking the wiring looks
right. Does any know what could cause this?

thanks