This sounded like an interesting alternative to blowing travel dough, so I just looked into it. It looked a bit more adventuresome that DW would appreciate. I was looking at Crescent / Sunset Ltd / Texas Eagle / Missouri River Runner / Southwest Chief to get to LA from Charlotte. The Amtrak website was pretty horrific, but with effort, I was able to power through.
The itinerary is pretty rough, starting out with a 2:55am departure. Arriving in New Orleans at 9pm. Three nights in N.O. then 9:00am departure, dumping you off in San Antonio at midnight. That's an 'ouch'. Find the hotel for that first night, plus an additional night in SA, then a nicely scheduled 7am to 7am trip to St Louis. And 4pm that same day, a quick 5 hour trip up to Kansas City, for a 9:40pm arrival and a 10:42pm departure for 2 nights on the Chief, arriving in LA at 8:00am. It's those uncomfortable times having to get from the train station to the hotel that puts me off. I even looked at the fancy Amtrak tourist booking site, although they book your hotel near the train station, they don't include transfers. So you're dragging your luggage around in the middle of the night trying to figure out how to get to the hotel. That just ain't going to fly with DW. The price was $4282 for bedrooms except business on the 5hr leg.
I think the solution for us might be to first get to a place where one train really starts, and stay on it until it ends, and that's where we'll end-up. The longest leg in the above itinerary is 35 hours from Kansas City to Los Angeles.