random disconnects from a single application

Nothing special from Cox but perhaps something special your old router was doing. First see if UPnP will do the trick. See here on how to enable that. If that doesn't work, disable it and use port forwarding instead. See here on how and here are the ports

1. Open outbound access for TCP ports - Second Life servers do not establish inbound TCP connections to client systems running the Second Life Viewer software. Instead, they use the "request / response" message pattern. Enable outbound TCP access for ports 53, 80, 443, 12043, 12046 and 21002.
2. Open outbound "session" access for UDP ports- Although UDP is a session-less transport, many firewalls block unsolicited incoming UDP traffic to a particular port unless it has seen recent outgoing UDP traffic from that same port. Activate outbound UDP for ports 53, 3478, 3479, 5060, 5062, and 12000-29999.

If you can't get that to work, see if there is a firewall setting you can adjust. Finally, you can always connect direct to the modem, reboot the modem and try a game then. If you get disconnects then, it's not a router issue.