|You really do not need to turn off a monitor since it powers down to standby when the computer is off or on sleep mode.|
If you are also leaving the computer on when you are not having trouble, it could be the video card that is the problem (or even your power supply). Test with the other monitor under all conditions to be sure.
Getting a dual output video card would work, but if the monitor is going the best answer would be replacing it.
You have to be a little bit crazy to keep you from going insane.