I'll try to understand this better, but I have a few questions.
1. The computer is not running, and is then switched off?
2. As long as the power supply has power going to it, the resolution doesn't go wonky?
3. Have you tried a different monitor cable?
4. Have you switched off the monitor as well?
5. Have you checked the BIOS battery?
As an aside, I recently had a problem with a DVI monitor (20" 16:10 HP TN panel) causing a no-resolution issue, and corrupting Win7. I'd roll back, rip and reinstall drivers in safe mode, reboot, and lose it again. I tried a different monitor cable, no change. I tried a different monitor, and the computer ran fine. So I hooked the questionable HP up as secondary..it tried to become the primary, and then caused a no resolution issue on itself, while the primary was just black (like my secondary was normally set up). I tried another HP of the same model, and the issue went away.
So, it could also be the monitor causing some funky issues. I've never had the above problem I mentioned before, or heard of it, but I believe if some pins in a monitor short out, or the circuitry is damaged/used up (components in that case), the monitor could potentially be the issue.
I hope some of this helps point you to the solution.