Have you tried a hard reboot of the router? Hold the reset button for 20 seconds and see if that helps. If possible, post any logs coming out of the router.
Edit: It's possible that your router needs a firmware update. I'm seeing stuff online about other verizon routers that the crashes were fixed after a firmware update. However, it seems like Verizon doesn't send out the firmware update unless you ask for it, and I don't know if its available online.
It seems the core of the problem might be an incredibly small NAT table. Open up the client, click options, and turn Peer to Peer off. This might help a bit, but I don't know if it'll completely solve your issues.