Thank you for your advice everyone.
I made a mistake when I said this. The route add problem has been resolved, but I believe that was only a symptom of the underlying problem, since even though I had the "-p" in the wrong place, it would still remember the last configuration.
I think my router is timing out and Win2003 cannot see it, so it reverts to the other router. Doing a release/renew wakes it back up for a few hours, then it times out again. And as a result, it screws up DNS resolution and all internet connectivity is lost for the entire LAN except for web pages that browsers already resolved, or applications that remember specific IPs.