How does the coax get from the street to the different rooms? Where is the amp? It should go from outside to a 2 way splitter, one end going into modem and one end going into amp, then from amp to the different boxes in the house. Not sure if the signal issue is related to the logs and packet loss, but it's not optimal.
What do you mean by "fiber connection"? Does Cox or someone else offer fiber(fttp) service in the area?
As for 2Gbps, if nothing else, you would need a new modem with a 2.5Gbps ethernet port like the Arris S33. You would also need a router with at least two 2.5Gbps ports, one for WAN(modem) and one for LAN(computer). If/when you upgrade, do it as a professional install to confirm everything is working while the tech is there.