First, welcome to OTOL!
That is quite a trip you have planned, going through virtually opposite corners of the USA (northwest and southeast).
Regarding Washington, DC, I would plan something, but nothing that you would need an appointment for or that you would pay for in advance. Yes Amtrak has gotten the reputation over the years of being late, but in the recent two years things have much improved and it's not strange for a train to reach its destination early!
. So you could end up with even more than 5-1/2 hours on your hands. That is not to say that your train might be a little or a lot late -- every trip is a new adventure.
The Washington Metro serves much of the city, and almost all of the sites are within walking distance of a Metro station. The Red Line provides frequent service to Washington Union Station. Just follow the signs to Metro after you arrive.
I believe that the White House tour usually has long lines, something you don't want with limited time in town. The Smithsonian may be a better idea, as you can choose among many different museums to suit your interests and go at your own pace. There is a Metro station on the Blue/Orange Lines at Smithsonian, but the complex is so enormous that some of the buildings are actually closer to other stations.
We have some members who live in northern Virginia, just outside the city of Washington, who may be able to add something about what would be a worthwhile and reasonable use of the dwell time you have between trains.
While it's wonderful to plan so far ahead, please keep in mind that Amtrak changes its timetables at least twice a year. We just had one the last full weekend in October, but there will be one more in the Spring, usually April or early May. If you do make reservations this far in advance, double check them after that time, as the long distance train schedules might change slightly.
Enjoy your trip, and after you go please let us know your impressions!