My sister took an NEC train from Washington to Philadelphia on a holiday weekend. It was sold out, and she ended up standing for a good portion of it.

There should be one seat for every passenger on the train, but the chances that everyone on the train winds up in the right car and finds that seat are slim. Hopefully as the trip progresses things will "equilibrate"; as was mentioned above, if you're traveling with your child, the crew and kind passengers will be accommodating.

Maybe it would help to figure out what platform the train is leaving from, and then insinuate yourself next to the doors/stairs before boarding begins.

As to tickets, prices usually climb as the date of travel approaches and more and more tickets are sold, BUT I myself can't say whether Amtrak sells last-minute tickets cheaply (to minimize empty/non-rev seats), or sticks to its guns, pricing-wise. Maybe one of the more knowledgeable posters on this board could answer that question.