High latency isn't really caused by a copper decommissioning. Typically, getting people off of copper and onto other technology like Fiber and Wireless would have the opposite effect. Verizon is probably correct, and this is a case of congestion. Are your speeds consistent when running a download? When downloading a large file or running a speed test, if you look at your task manager, is the graph for Ethernet/Wi-Fi consistent, or is it varied? When performing this test, make sure no other devices that connect to your network, Wi-Fi or Ethernet, are on and using data.
Another data point - does this continue to happen after 12AM, or before 9AM? These are normally the lowest usage hours for Internet service. If service is consistent during these hours, it's certainly congestion.
Depending on how your area is configured, Verizon may be able to move your connection through a different path in the network. A network tech will be able to take a look and help sort this out. However, if the problem is at your DSLAM (where your DSL signal originates from), they may not have as much to work with. There may not be an extra DSLAM or other circuits running into the DSLAM to help ease the congestion.