I had this happen to me when I was booking my TE trip a couple of weeks ago.
I had my laptop at work, was checking prices OK. I hibernated the laptop, went home, and then kept getting 500 errors. Thinking there may be site maintenance, I waited until the next morning (it was hard to wait, let me tell you - I was ready to book). Anyway, I was still having problems the next morning - same 500 error (500 is generally known to be an "internal server error" for those of you that don't know and usually says something at the server is a problem). Anyway, I rebooted before work, as I always do, and then I had no problem!
So I'm wondering that if those that are having issues have moved from one network (or IP address?) to another. If so, try a reboot and see if that helps.
Yes, I know that's "standard help desk talk", but it seemed to work for me and I'm wondering if Amtrak's server may be storing session state based on something network related and gets confused when a client machine switches networks.