Hi Ordindrake. You could be running into a couple of different problems.
First, are you giving the system some time to register the changed passwords? It takes around 15 minutes for the system to register the change, so if you change passwords a lot the system may not catch up and get confused. Change your password one last time, give it twenty minutes for good measure and try logging in again.
If that doesn't work, check the very bottom of the first client window and check what region your client is set to. It's the very bottom right hand corner, and it's pretty small. Make sure it's set to the US. That happened to me when I first downloaded the game a million or so years ago. I had registered a US account, but for some reason my client was set to EU/Spain. As soon as I set it to US, I was good to go.
If neither of those things fix your problem let me know.
Good luck