I had exactly the problem that you describe with my 7 yr old computer - not much memory and Win98.
I spoke to several so-called experts who diagnosed a faulty monitor and / or a faulty graphics card. Both of these explanations seemed feasible as I had been - and still am - having problems with the monitor brightness randomly changing from normal to darker and back. This suggested a faulty switch or connection somewhere along the line but I couldn't find anything amiss.
In the end, I started browsing through catalogues with a view to actually lashing out on a brand new shiny replacement computer. I shut everything down via the mains switch and went to the pub.
This is my usual solution for all my problems and it worked!
When I switched on next morning everything was back to normal apart from the computer auto-running scan drive which it does when the system hasn't been shut down properly.
I still have the variable brightness problem so there is something slightly amiss but apart from that - no problems.
I am putting the problem down to one of life's little mysteries.
Incidentally, have you tried holding the Power switch in for 5 or 6 seconds when you can't switch off?
Sorry for the rambling - hope it helps - or at least gives you some comfort.