When the FSR fails, the symptoms are usually that the blower motor doesn't function correctly. It only controls the interior fan motor, and it sounds like your fan was working fine.

Problems with not getting cold air out the vents can include low refrigerant, the auxiliary fan, or the control valve.

Go ahead with the FSR, and if it helps, great. If not, it is time for a proper diagnosis before you continue to replace components.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue