How about breaking up the journey for a few days somewhere along the route.
I travelled from Chicago to Dallas last year with a three day stopover in St Louis. Missouri.
From St Louis, you can take the Missouri River Runner to Kansas City, where there is a connection to the South West Chief.
Or you could travel from Chicago to Los Angeles on the Texas Eagle, which runs via St Louis, Dallas, Austin, San Antonio and El Paso.
The South West Chief makes several stops en route, but you have to watch the time of day. Also, not all Amtrak stations have taxis and public transport nearby, especially in the middle of the night.
In Chicago, you could also take the South Shore line out along the Indiana Dunes. The route through Michigan City, Indiana takes a full size train right down the middle of the street. Search for "11th Street Station Michigan City, Indiana" for pictures.
The superliner roomette is ideal for a single person, with the proviso that you don't have too much luggage. If you do, you can check it in like you do on an aircraft.
All meals are included, but alcoholic drinks are extra, and as elsewhere in the US it is advisable to leave a tip.