As I near completing smaller details, I ran into a possible snag.
According to Renfe and Trainline, all trains in the morning are booked (May 25th - 28th).
If not able to train there, what is the next best option (Car?)?
Bus. We rode the train to Segovia, but actually took the bus home (better schedule).
Avanza runs buses from Madrid-Moncloa to Segovia. Scheduled travel time is a bit less than 1-1/2 hours. I believe there's a Metro station at Moncloa, but it might be a bit of a walk to the bus departure point. You'll need to do some research to figure out how much travel time to allow.
It appears tickets are on sale until the end of April, so you'll need to wait a bit, but I doubt the schedule will change much between April and May.
Nick must surely be right. "The requested train is not available at the moment..." doesn't mean it's sold out. On the Renfe website, sold-out trains are usually marked "Full".
The bus will work, but the train takes about 1/3 as much time.