Abandoning ship would be remarkably atypical for a starship crew. In comparable situations, Kirk's crew never budged: evacuation would only happen as per the Captain's explicit orders. The two times we witnessed a starship being evacuated, "The Doomsday Machine" and "This Side of Paradise", it happened with astonishing discipline. In the first, all but one of the crew beamed down to what they probably knew was their death, within a short time and without a hint of panic. In the second, the crew had supposedly lost its respect for authority, and still filed down neatly and efficiently.
Sending a distress call at the first sign of trouble is another thing starships apparently never do. For all we know, it's a hopeless endeavor anyway: starships might operate outside communications range for most of the time, and the only way to report back to base would be to launch a recorder marker. No doubt the Exeter crew would have done that eventually, but we have seen how much it takes time to prepare one.
On the other hand, it's hard for me to imagine Captain Smith thinking: "Yes, we've stuck an iceberg and we're going down by the head. But why even bother sending out an S.O.S. After all, we're just so darned far from our base. Also, we're the fastest ship out here--there's probably not an Olympic-class vessel near us, and who would come to our aid--a freighter? It's clearly just not worth the effort to fire up the Marconi."