What you describe sounds like a failing graphics card. My particular troubleshooting process is to swap in another one to see if it works. It's handy to have a few extra parts. Short of that, you might try downloading ATIccelerator and using it to downclock the card. Downclock both the GPU and memory by 5 to 10%. If it helps, that would give a more clear indication.
Also, if downclocking helps, it might still be possible to salvage the card. As with virtually everything else, Japamac has a nice tutorial on the subject of card maintenance:
First read his overview:
Then read the teardown instructions for your specific card:
Applying fresh thermal compound is not an optional step. Maybe wait to see what others have to say about this suggestion before purchasing the compound. The fact that it isn't doing it all of the time leads me to believe that it might not be too late, but I don't know how common it is to bring them back once they start showing problems.
Before anything, I might try reseating the card and not just wiggling it from the outside. Take it out, blow out the dust with some compressed air, and verify that the fan moves freely. Clean the contacts on the card with a pencil eraser. Also verify that the fan is actually working when you put it back in.
There might be something else obvious to try that totally eludes me at the moment. Hopefully, you'll get some better feedback than this. Good luck.
Swampus - thanks for the reply. Sorry it took some time to get back to you (computer problems, you know ...). I tried downclocking, which had limited, if any, effect. I found an inexpensive used video board. It arrived today but wouldn't you know that this morning (before the new video board arrived) the computer would not wake from sleep. After forced shutdown, on restart the fans and drive spin up, but no chime and no video. I put the new video board in, but no difference (although its fan is much quieter than the old one). I then noticed that the power light on the front of the case is on only when I'm pushing the button, otherwise it's out (though the fans are spinning). Also, the usb never comes on (i.e., usb peripherals get no power). I swapped the battery, did a PMU reset (with the internal button) tried various combinations with the memory I have, used the heat gun trick. Nothing - can't get the power light to stay on even though there seems to be power.
Any further ideas would be welcome. Thanks again.
What a bummer! Have you had a chance to test the PSU yet?
If this is a machine that you'd like to keep going a while longer, I'd keep an eye out for a working parts machine. In my area, working dual 2GHz models can be picked up for $100. That would give you an extra PSU, motherboard, CPUs, video card, and extra ram that you can start using right away. It takes the guesswork out of troubleshooting when you can swap in known good parts.
Please keep us posted.