I'd say that the rhyme was simply a general policy, and not any sort of actual danger.
In "The Squire of Gothos," for example, when escaping from Gothos, the Enterprise is moving at full-warp when Kirk orders hard to port and starboard in order to avoid the planet Gothos, which Trelane is moving into the ship's path.
Likewise in "Journey to Babel," the Orion-attack ship is clearly maneuvering laterally in its attack-pattern at high warp, otherwise its higher speed would not give it an advantage since the motion would be relative. But the ship doesn't blow up from such maneuvers.
Recall that the full line is "Faster than light, no left or right: course-corrections can fracture the hull."
The key word here is can, i.e. it's possible but not certain. Thus, in conjunction with these and other examples, we see that it's simply not a good idea when it can be avoided. But that aside, we've clearly seen the ships maneuver at warp on various occasions when necessary, even at warp 10 (TOS scale) without damage.