If it says network cable unplugged, it's almost certainly a hardware issue. This indicates that the network port is either not detecting link state correctly, or is in fact not getting link at all, both of which point to physical issues with the ethernet transceiver or cable (you replaced the modem, so we know its not likely that).
If the cable and/or port was good, you'd see errors indicating limited or no internet connectivity, which means you have link, but not a routable IP address.
You could try going into the properties of the network card (Control Panel -> Network and Sharing -> Change Adapter Settings -> right click on network connection -> Properties -> Configure, and then look for link speed and duplex) and setting the link speed/duplex to 10 megabits half duplex, which is the lowest common denominator setting, and see if that helps. If the transceiver or cable is flaky, often this will at least get you working again until you get it fixed.
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-Scott
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