As I've seen lately that Verizon is keeping Internet on the Ethernet leg only depending on speed and since you already have it installed that way, it is likely that you'll stay internet over Ethernet (unless you really want to force the issue).
So, in this case ... install the new router in place of your old router with no changes. It'll work just like the old one. Next, connect coax to your ONT and then use properly rated splitters to split the signal to go to each STB and to the router. The router will bridge the ethernet between your local wired ethernet/wireless and the coax network to give the STB's a path to the internet.
If you decide to remove the Ethernet for Internet (or Verizon decides to switch you), once you have the setup in place above and working, you would need only to disconnect the ethernet cable to the ONT and have Verizon make the switch the internet of coax (and potentially a router reboot which may require a little Verizion assistance to insure your MAC is released so you can DHCP a new internet address). The router will handle both internet over Coax and local LAN over cox simultaneously (technically called MoCA, it will use two different frequencies on the coax to manage the signals).