Am I correct in my understanding of the rules governing courtesy runners - that the same courtesy runner may not be used for both the pitcher and the catcher at any time during a game? And if so, how do you keep up with this as an umpire? ( Score card, line-ups, or elsewhere?)

Speaking ASA, you are correct.

I make a notation on the lineup card "CR-P" or "CR-C" next to the CR's name.

That is correct for Fed also, but not USSSA.

USSSA does not allow the same CR to run for F1 and F2 in the same inning. That would alow it if it occures in different innings.

USSSA also allows any player not currently in the game to be a CR, while Fed only allows a player who has not participated in the game to be the CR.

I have a line on my "game card" that allows me to record the CR for each position during the inning they run. (The card also has a line for defensive and offensive conferences as well as the score for each inning, and the # of the pitcher and catcher of record. Very helpful if there is a suspended game to be resumed later, even if different umpires get assigned to finish the game.)

Roger Greene

I do basically the same, but just make notes on the teams' lineup cards. I don't have a score card or an elsewhere, but RG's game card sounds good.
The key is being alert enough to be sure the coach reports the change.