Thanks for the tip on currentReadTime! That part works now without caching the read time.
But, unfortunately, the problem still persists. I have a little debug script that gets all Audio Sources in the scene (via FindObjectsOfType) and prints what they're playing — so I can see that the STM's Audio Source is playing while the characters are talking normally, but not after Continue(). For one reason or another, it seems like the Audio Source just isn't playing even though the text is reading.