When he discusses doubled thirds he means specifically the third of the triad, reason being that it's the most conspicuous constituent tone in the triad and therefore the least in need of reinforcement by doubling. (Page 36 of my paperback copy.) So it's not about thirds in general.
I don't remember anything about the sustained thirds, though, but I'm intermittently going through these sections myself so maybe I'll run into it at some point.
Edit: I see you gave a page number, and I looked it up. I think what he means is that if you run into a situation where you're sustaining a doubled tone (which isn't the third) into the next chord (in which it is the third), you should see if you can make the doubling note move instead of sustaining.
Second edit: Look on page 72, the third and fourth example of I6 - III.