Delvo, you probably can't find anything more specific on Albinoni's Adagio (also known as the theme from Orson Welles's The Trial) because its authenticity is somewhat in doubt: though it purports to be based on a fragment by Albinoni, it's basically a modern piece. Still nice, but the work from which it was excerpted is lost.
For sad and haunting string works, there are some good possibilities in string quartets, or at least in portions of them. Not all of every quartet will have what you're looking for, but some of the best stuff there is can be found there if you look.
Schubert's "Death and the Maiden" might be a good start. Try Smetana's quartet subtitled "In My Life," also. This is a very intense and tragic work by someone who really knew how to wring out a melody. Janacek's first quartet, subtitled "The Kreutzer Sonata," is quite intense as well, and highly recommended. If you ever saw the movie of The Unbearable Lightness of Being, you will have heard some of it. For a little extra dose of lyrical strangeness, you might also hunt up a recording of Shostakovich's second piano trio (that's piano, cello and violin).
Barber's Adagio has been mentioned. I don't care much for the orchestral version, which strikes me as overwrought, but it was originally a movement from Barber's string quartet, and in that form I think it is much more effective; the small string ensemble gives it a bit more edge and emotional immediacy, and saves it from being so lush and mawkish. If you've never heard it in this form, you've missed something.
