Hi and welcome to ExTS
If it gets to the desktop when cold - then afterwards shuts down sooner it does sound like overheating.
Your loose heatsink wouldn't help - but that, presumably, has been rectified.
The graphics problem is a cause for concern as most m/boards have the graphics in-built.
If you have a separate card you are lucky.
Fitting the new fan should solve the over-heating problem but I fear that you still have a problem with the motherboard.