Just so that I'm clear, I'll try to reiterate what you're saying.
1. You have an
AverMedia HD DVR capture card.
2. You want to record continuously for long periods of time, but...
3. When your monitor turns off/computer sleeps/screen saver comes on, recordings stop.
4. Your saving money by not renting the cable companies DVR.
Hopefully I got that all right.
I need to understand further on point #3 first. Do I have it correct that your recordings stop when either your monitor turns off or the screen saver comes on? Also, are you using AverMedia's software to record? The reason why I'm curious about this one first is because 7 years ago when I first started recording, I was using the supplied software with my tuner (from Hauppauge) and whenever the monitor turned off or the computer went to sleep, my recordings suddenly ended. So I always had to leave my monitor on. It's likely AverMedia's software that is at fault and nothing at all to do with your computer. I would have suggested trying to use your card with Windows Media Center, but the AverMedia site says that it will not work.
If leaving your monitor on for long periods of time is an option for you, you should go to your Power Options and set sleep and display auto-off to Never, if you haven't already done that. As for turning the screen saver on/off at will, I'm not aware of a simple way to do that aside from how you're doing it now.
OK, now back to points 1, 2 and 4. I can't speak to this particular capture card (although I'm sure it is very similar to the old Hauppauge HD-PVR) so I'll explain how I record instead. Currently I'm using the
SiliconDust HDHomeRun Prime which requires a CableCard from the cable company. I use Windows Media Center to record, which comes with Windows 7. Windows Media Center includes a TV guide (up to the next 14 days) that allows you to pick and choose what you want to record. You can set it to automatically record full series weekly/daily or individual shows at specific times, whatever you want to do. Hell, you can manually set it to record a single channel continuously for 23 hours and 59 minutes (the longest you can set) if you really want to.
I've had my computer on nearly everday for almost 5 years now. I record all day, every day, three HD channels at a time on Windows Media Center with no trouble at all. And believe me, my PC is nothing special. Doing what you're trying to do is definitely possible, but I'm not sure your card or the software that came with it is up to the task given the trouble it's causing you when your PC is performing minor routines.
Sorry I couldn't be of more help, but hopefully this might give you some direction on other recording devices and software.