Have you set the on-board nic in yast>network to not configured?
Also, if you are using dhcp via the router, check the lease time or else just set up with a fixed ip address.
I have a similar problem and also have 2 nics, but have never had any problems.
You could try setting the onboard card to never be activated in Yast, Network Devices, Network Settings (your bios may also have an option to disable onboard lan)
Me if I had 2 cards, 2 os’s and each os ‘liked’ a different card, I’d use two cables, and just let each os use the card it works best with, at least until I could figure out why the connection was dropping
You don’t mention why the onboard card doesn’t work in windows, if you have a driver and it still doesn’t work it could be that the onboard one is faulty, which I suppose could be the root of your problem in the first place