Welcome, of course it's OK to ask questions!!
I think the fan you hear running after you turn off the car might be the climate control "sniffer" fan. I assume it is rather quiet, not nearly as loud or volume of air as the blower motor? If so, this is a small fan inside the dash at the climate control panel. The supposed philosophy is that this fan runs after shutting of the car, and occasionally when it is hot in the car, to keep the air around the inside temperature sensor accurate, so that when you get in the car, the heat or air conditioning is not running full blast. I've never seen this officially discussed though.
It is normal for this to run for a few minutes after shutting down, but not continuously, killing the battery. This does not have anything to do with the final stage.
If it IS the main blower motor, then it would most likely be the final stage resistor. But they usually fail the other way - loss of one or all speeds (no air at all). I guess it could also fail in the ON position?
There was a known software glitch on some 3 series 2003 - 2004 climate control panels. When they glitched, they stayed partially powered on and would kill the battery. A software patch fixed this problem. I wonder if that might be the case for you?
Try to isolate exactly where the noise is coming from - the climate control dash panel or the main blower?