Power on and off issues

I've been having odd power up and power down issues. When I press the power button the LED's and fans come on for a few seconds then it goes dead, when I press the power button again it starts up normally(**). When I try and shut down the computer it goes through the shutdown process but doesn't power off - so it just sits there with the windows shutdown screen - this happens when I ask it to shut doen OR restart so I have to manually switch the power off.

** If I press and hold the power button the power will go off but I have to press the power button twice to get it to start up again BUT if I switch the PSU off then on again I only need to press the power button once to get the computer to start.

The real problem is that windows doesn't seem to recognise that it has been restarted so every time I switch off it installs a long list of automatic updates but when I switch on again it tells me that the same updates are ready to be installed. I've now got some software that won't run because it says the computer needs to be restarted - even after restarting.

I've tried a new PSU but it didn't solve the problem and I updated the BIOS. I haven't installed any new hardware for several years, everything has worked fine for years.

Could this be a motherboard problem, and is there any way to check (without buying a new board)

The motherboard is an Intel DP35DP.

Any help/suggestions would be appreciated.