Need the IP for your game server, not the
www.eidos.com webserver. That server is in a subnet managed by a different company, and because it's physically in a different section of Montreal, it breaks off differently once it hits the Montreal area. Open Resource Monitor (run resmon) and find FFXIV in the networking section to see what IP's you are connecting to when in game. There will be more than one. You need the one starting with 199 for NA/EU (the one starting with 124 is a connection to Japan and is idle most the time).
You likely will still come across those struggling cogentco segments though...they've been having problems in both LA and San Jose for some time now. Interesting that it is following over to Texas too.