Well, I don't think there were any recognizable American destroyers in "Run Silent, Run Deep." (The distant shots were all of models, if I remember; there were a few middle-distance shots of depth charges being dropped, etc. - clearly made on board American ships - but they were too generic to identify the ships. In any case, that movie was in black-and-white, so it wouldn't qualify.
I've seen "The Key" only once. The story centers around British salvage tugs; there are some nice shots of such vessels, and of "damaged" merchantmen - and British submarines masquerading (not very successfully) as U-boats. But, so far as I can remember, no American destroyers. And that one's in black-and-white too.
My wife and I watched "South Pacific" on TCM the other night; I don't remember any ships that were recognizable.
The one I'm thinking of isn't particularly well-known, and I last saw it quite a few years ago. My assertion that the ship, on board which most of the action takes place, is a Benson/Gleaves/Livermore-class one is based on my memory - and it's probably stupid of me to rely on it to this extent. But I'm pretty sure I'm right.
The date 1958 is from Leonard Maltin's 2009 Movie Guide. The information in the book is completely consistent with my recollection of the movie - though Mr. Maltin, of course, tells us nothing about the ship.
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