I have taken the Texas Eagle/Sunset Limited from Dallas to LA and back several times in the through cars.
To the best of my recollection of my last trip from LA a couple of months ago, the consists was:
Two locomotives
Baggage
Transition sleeper
Florida bound sleeper
Diner
Sightseeing Lounge
Two Florida bound coaches
Chicago bound coach
Chicago bound sleeper
When the train arrives in San Antonio, the Chicago coach and sleeper (train 422) are uncoupled from the rear of the Sunset (train 2) and coupled to the rear of the waiting Texas Eagle (train 22).
The Sunset is scheduled to arrive in San Antonio fairly early in the morning, the Texas Eagle isn't scheduled to leave until around 8:00 am or so. Except for recently it seems, there is usually enough time for the switching to be accomplished and for the Eagle to depart reasonably close to on-time (for Amtrak) or least soon enough to gain time through schedule padding farther up the line.
Heading back to LA, the Texas Eagle (train 21) arrives in San Antonio the night before, with the LA coach and sleeper (train 421) at the rear of the train. At some point during the night, thes two cars are removed from the Eagle and coupled to the rear of the Sunset Limited (train 1) once it arrives in San Antonio early in the morning.
I've been lucky on my trips... Eagle and Sunset haven't been so late that the connection wasn't made. Although, the Eagle has been held for several hours on occasion waiting for the Sunset to arrive from LA. And one trip the Sunset was so late, that we actually connected with the Texas Eagle that departed day after we were scheduled to connect. That wasn't really related to the Sunset's poor on-time performance, but rather due to a consist not being available to leave LA on time because of a problem with the Southwest Chief. The Sunset was scheduled to depart LA at 10:30pm, we left at noon the following day.
Enjoy your trip and just be prepared to go with the flow no matter what happens.