So it seems that I solved my problem, although I'm not sure which update caused the problem to disappear.
The first thing I did was update to the new F7 BIOS, loaded optimized defaults, then set all the wake-on-LAN settings off. I also downloaded the built-in motherboard LAN drivers directly from Realtek's site. I noticed the drivers from gigabyte were dated in Device Manager as over a year old. After installing the new drivers from Realtek, I set all the wake-on-LAN settings to disabled in the driver. Both the BIOS and driver settings were the same as before, so clearly an update did the trick.
Looks like everything is working well. I had the PC in sleep mode for over 2 hours yesterday, which I was not able to do before.