If PC is still under warranty, contact Dell Tech Support. Have Service Tag handy.
Running BIST on a PC without a BIST button, like the XPS 8940, doesn't mention anything about the PSU fan, like it does for PCs with that button, so that's an unanswered question about health of your PSU.
If you're using a power strip, surge protector, or UPS, remove those and connect PC directly to a working outlet.
Is the LED on motherboard on 24/7 whenever PC is connected to a working outlet?
Have you checked the power button connector to motherboard?
It's possible the power button module is the problem, so read this thread.
This model of PSU only has the green LED no BIST button beside the light. I currently have the side and front panels removed. In addition, I removed the RAM modules and the HDD. The power button does not seem sticky (it makes a sharp clicking sound when pressed).