I finally got around to installing my own wireless router on my FIOS network, a Linksys e2100L with dd-wrt installed on it. After the router is setup (that's beyond the scope of this post) there are 2 tricks to make this work.
First, Verizon FIOS gives out really long dhcp leases, and doesn't want to give them up. So you need to not only clone the MAC address for your router, but actually set it to the ip addresses that your old router was given. I'm told that after about 2 weeks you'll be able to start using DHCP again, but you can't for the switch over.
Secondly, you have to set the upstream MTU. Presumably Verizon is doing some VLAN tagging, which would explain why your IP addresses can jump all over the place after a major network change on their side. 1496 should be a safe value, and it looked like it worked, but I left mine down at 1450, for no good reason other than superstition. This was the trick I was missing before, and since I've been dealing with bizarre networking issues at work recently the idea was still floating around in my brain.
It's now working, and my port forwarding is setup enough that I can do any fixes I need remotely via vpn.