Ethernet and Wireless LAN only work when laptop is on AC power

I just picked up a used Sony VAIO VPC-EG37FM/W laptop. I was aware that it had some issues, so I got it real cheap with the intent of fixing it up.

As I started working on it, I seemed to experience very erratic, random issues with the network connectivity. After several reinstallations of Windows 7 and Sony drivers, however, I have identified the pattern. When the laptop is on AC power, connectivity is not a problem. When I'm on battery power, however, the network adapters cease to work and actually completely disappear from Device Manager. The is true even when the battery is fully charged and everything else works normally.

I have already tried going into all the power management settings for the laptop itself and for the network adapters and made sure that they are not being shut down when the laptop is running on battery power, and yet the problem persists.

Any thoughts? Thanks!