My data is from over a half decade ago, so could have changed since. Amtrak worked on a bucket approach for ticket pricing on the long distance trains, the first bucket of tickets were cheap, when they sold out the next batch of tickets cost more, but when sold out the third bucket of tickets cost even more. That way if people were willing to adjust their travel plans, they were enticed with savings to switch to trains that were less full. I can't imagine that you could buy a cheap ticket and then show up on a busy day to claim a seat.

When I and a friend rode Amtrak from near Chicago to Portland OR six years ago to start our Pacific Coast tour, by changing our travel date by a few days were got a Roomette for about half the price of other days, so we changed plans and got the cheap Roomette instead of riding in a coach seat for a long time.

When three of use rode from DC to Chicago several years ago, we bought all three tickets at the same time, two of the three tickets cost more, in other words the first ticket we bought emptied that bucket so we started buying out of the next bucket.