Would your active Century account hinder a Cox account? No. Your new modem would "point" towards a different provider. With your Century modem disconnected, Century would have nothing to communicate with in your house. If Century can't establish a link to your Century modem, no communications could occur. If no communications, no data traffic from Century.
The USB outlet Century set up in your den sounds peculiar. I think, perhaps, Century rerouted the primary coax in your house.
First, I'd locate how "the grid" connects to the exterior of your house. A coax cable could emerge from the ground or connect from a telephone phone. A good indicator would be a little metal or plastic box attached to the side of your house.
Second, I'd remove the cover plate of the coax outlet inside your house closest to this exterior connection.
Third, you want the exterior coax connected to this coax outlet. If it's not connected, this may be your problem.
We can explain more later but in the meantime, is this coax outlet connected?
Note: Little metal or plastic boxes outside your house are called Network Interface Devices (NID).