I work a lot as an ASM. Once I get into the final phase in the rehearsal hall (think designer run-throughs), I usually abandon my script in favor of run sheets. A script is too burdensome once I'm trying to run the deck and truthfully, isn't all too helpful. A lot of transitions, QCs and tech elements that I manage as an ASM happen in the blank parts of the script, between dialogue / scenes, so why continue to use that document as the primary reference point?
In tech, I keep a copy of every department's run sheet, plus the entrance-exit chart, for easy reference of "what's next." For my own runsheet, I just print a copy of the running order with lots of space for each scene / transition, so I can quickly jot in my track as we tech.
Once we're in previews / open, I'll convert that handwritten run sheet into a more formal, "pretty" runsheet that can easily be handed off to another SM if necessary.