Hi Imhotep,
Welcome to PCHF.
You could try using a 3 - 4 pin adaptor, and plug your fan into the power supply instead of the motherboard. This way, the power and speed is not controlled by the motherboard, and will run at a constant speed.
3 Pin to 4 Pin Fan Adapter with Separate RPM Sensor Sleeved [3TO4FANRPM] : Performance-PCs.com, ... sleeve it and they will come
However, this is likely to be a more serious problem than just your fan. The fan will usually continue spinning, even if the
CPU temperature is low. If your BIOS (Press F1, F2 or [DEL]) has a section called "Smart Fan Control" or "QFan", make sure it is disabled.
Otherwise your problems are more likely to be either a motherboard (short), or power supply. You said you could understand why this would happen to a new PC, but not a PC you have been using for years? Your answer... Capacitor aging! Your power supply will degrade as time goes by, and as a result will consume more and more power, leaving less for your PC. If your PC is being starved of power as a result, it may well explain your problems. Have a look at
eXtreme Power Supply Calculator v2.5 (click on Go Lite), and fill in the specifications of your PC. Be sure to add 15%-20% for capacitor againg. Compare the results with your current power supply.