MacBook Pro - Cannot sign in to Apple ID - server error

I have a 15" MacBook Pro that is two years old. I paid over $2,000 for this machine because I added RAM and increased the hard drive capacity over the stock model. The "S" key stopped working on my keyboard so I went to the Apple Store and they told me it would cost $800 to install a new keyboard. I did not want to invest that kind of money in the machine because I am becoming disenchanted with the quality of the past two Macs I have had. The prior Mac was a standard MacBook and after having that machine only two years the display became distorted. I checked that problem online and found it to be a known issue with the model MacBook I had. I took the machine to the Apple Store and they refused to repair it because there was a small scratch on the outside cover of the laptop and they claimed that the scratch must have put pressure on the screen and caused my issue. Really??? So I bought this machine.

Here is my current problem. My laptop password happens to contain two letters "S". Since I didn't want to spend $800 on a new keyboard the Apple Store sold me an external keyboard for $100. I took that home and, for a few days, I was able to sign in to my MacBook Pro. Then it started shaking its head every time I entered my password. I tried using the external keyboard in both modes - wireless and through a USB port - all to no avail. My machine just keeps shaking its head. I selected the option to change my laptop password using my Apple ID. When I enter the ID and password I consistently get a message saying there was a server error - try again later. Well I've been trying for three week now. I signed into my Apple account on another computer and the password works just fine there. I am at the end of my rope. I don't know what else to do. Can anyone help me?