If the fan only starts to spin fast once the displays become disconnected, then I do not think its an overheating issue, otherwise the fans would be the first early indicators of a impending failure.
When you say "crash", I assume you mean having an abrupt reboot. Given your symptoms I would subscribe that your failure order starts with the display output completely failing while your GFX CPU fan spins full blast, at which point either your computer has already rebooted at the first point symptom of failure, or it does indeed wait 3 or so seconds before the reboot.
Does your GFX CPU fan rev up whenever you turn on your computer? If so, then its simply possible that your computer simply rebooted by itself, causing all those symptoms by nature of the POST of the graphics card.
Now if its unusual in normal events to hear that fan go full force, thats a whole other story. If this is the case, there is a chance that the temperature detection circuitry is faulty, causing bad readings. I have seen this happen on at least one computer I worked on, but it was on the motherboard instead. Beyond that, shorts or other on-board faults of the GFX card could cause the GFX CPU to issue a reboot command by way of the internal watchdog timer (in other words, its waiting for a response from something that is taking way too long and shouldn't be) causing it to issue a reboot on the GFX CPU, and then would cause the driver to become desyncronized with the mapping of the peripheral port, leading to the MOBO CPU to issue a fault which, depending on the severity of the drivers condition, leads to a full abrupt reboot instead of the usual memory dump procedure of windows.
There can be other things wrong, but if you find the fan spinning up unusual, even on computer power-up, then something seems fundamentally wrong with a circuitry. Heat would not do this (and if it was, you would notice it!), and if it was a loose heat sink with the GPX CPU, then the crashes would happen nearly at the same time after boot-up, rather than randomly. If you think it might be a loose heat sink or not enough heat paste, if possible check that out on the graphics card just to be sure that you ruled out that possibility.
Ultimately you need to do a few more tests to figure out what the true cause of the problem is. If you had another graphics card to test with that would be super helpful.