Do Amtrak prices vary depending on how far in advance you book them, or are they the same price up until the day of travel?
Yes, it is worth purchasing Amtrak tickets as early as possible.
Amtrak does practice yield management, although it is not nearly as sophisticated as the airlines in this regard. Fares will increase closer to the date and time of travel.
If someone purchased a ticket in a low fare bucket and then cancels it close to the departure, sometimes the seat gets released for the original price; however, this is not something you can count on.
As of August 15, 2016, there is a refund fee of 20%. As a result of the new policy, the Refund Fee has changed from 10% to 20%. Reserved Coach or Acela Business now require cancellation 48 hours prior to departure for a full refund instead of the current 24 hours. Tour Passes now require cancellation at least two (2) days prior to scheduled departure of the initial trip for an 80 percent refund. ...
Unless stated otherwise, if a ticket is refundable, the passenger may avoid any refund fee by accepting the full refund value of the ticket as an eVoucher or other exchange credit, good toward future travel within one year.
If you are planning on traveling first class (a.k.a. sleeper service) on a long-distance train, you want to get your tickets as early as possible as the various sleeping arrangements (roomettes, bedrooms, family bedrooms and accessible bedrooms) often sell out, particularly in the summer. As the rooms become scarcer, the prices go up. Note that with the price of a room, all meals are complimentary.