I do exactly what you are trying to accomplish. First thing you want to do is get the new router connected and make sure you're online. Do your DHCP release on your Actiontec then immediately take the ethernet WAN and connect it to your new router's WAN. I have found the set-top box work better on the 192.168.1.1 subnet. I would recommend configuring your new router with that same subnet.
Next reconfigure your Actiontec before you connect it to your new router. Change your Actiontec to 192.168.1.2 and disable the DHCP server. I personally connect LAN of my router to the LAN of the Verizon router. But I have seen it done both ways. And that is really it. Reboot you set-top boxes and verify you have Guide and OnDemand.
If you use a MoCA adapter instead it's even easier. Just connect it to the coax from the Actiontec and connect the ethernet to the LAN of you new router.