Luckylabels

Hello All,
I recently upgraded my laptop to Windows 10 from Windows 8.1. After that, my Realtek PCIe GBE Family Controller (ethernet) randomly stops working. After a reboot or 'fixing' the problem, the card works for anywhere from 1 minute to 4 hours, then fails again.
Sometimes 'failing' means that in control panel it shows that it is still connected, but doesn't actually work, (http://gyazo.com/5851e6acdc0b526947cc1e94b0038d12),
or other times it just shows that it is unplugged, when that is certainly not the case (http://gyazo.com/d154deef358776847419324d6a0306e3).
Troubleshooting the problem with Windows troubleshooter yields three errors, seemingly random when they each appear.
The three errors are 'The Default Gateway is not Available', this error (http://gyazo.com/fd97077f10eaa6a637be32da61b94c3b), and then finally a DNS error that shows up on Chrome and the Windows troubleshooter (http://gyazo.com/4f80c18c329d4e10001d59a11db2bf58). The way the internet stops working is Chrome will stop loading web pages and give me a DNS error, shown in the most recent gyazo.
So far I have tried rebooting numerous times, (which USUALLY fixes it for a few minutes), troubleshooting with Windows troubleshooter, and changing the DNS servers to Google's (which I have now changed back to the automatic setting since my ISP, Time Warner Cable, isn't too happy about working on an external DNS server. I have also tried eight different ethernet cables, over two different networks and routers. All of this has begun after I upgraded to Windows 10.
I have tried three different drivers for the ethernet card; one from Realtek's website, one from my laptop manufacturer's website (Sager), and lastly Window's generic one, none of which have worked for me. It could be a coincidence, but it seems that if I have constant load on my ethernet card, it lasts for much longer without failing.
For example, I ran a test where I opened a live stream from twitch.tv and ran that for an extended amount of time. I have tried that 3 times, and every single time the card has not failed, with over 12 total hours of live stream watching in total now. Could be a coincidence, but I highly doubt it as the card usually fails within a few minutes otherwise. All replies are appreciated as I am extremely lost with this issue.
- Atte