Do I have to run an extra wire all the way to my new thermostat, or just to the "router"?
Trace where the wires from the thermostat to your furnace. You'll find that one connects to a relay and the other connects to a terminal on a transformer. R is the wire from the transformer, W is the wire to the relay. (Your thermostat may be connected to the wire backwards, but that doesn't matter.)

Your new C wire will be the other terminal on the transformer. Or if the transformer has multiple taps/terminals, then it will likely be labeled COM or -.

I suggest instead of trying to monkey with the wire to your existing thermostat that you just investigate where it's connected to and make new connections. That way if you run in to problems you can simply reconnect the old thermostat easily.