Ok. To answer your question about using another router, in most cases you'd have to get Verizon to change the Internet output from your ONT from coax to Ethernet. You'd then have coax feeding your TV(s) and Ethernet feeding your router.
The Actiontec router is a MoCA device, meaning it can be fed by coax or Ethernet. Most other off-the-shelf routers require an Ethernet feed to their WAN port.
Having said that, the troubleshooting you did seems to confirm that the problem you're having is not the wiring in your house. Have you told tech support of that step you took?
edit: What letter suffix is after the Actiontec's MI424WR model number? There's a wide range of suffixs that determine the vintage and capabilty of those routers. The latest model at the moment is the MI424WR Rev. I.